# Complete Centers, Normality and Singleton-Conjugacy Equivalences

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261115-r02-finite-group-center-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, No, No, No]], [2, [Yes, Yes, Yes, No, No, No]], [3, [Yes, No, No, Yes, No, No]], [4, [Yes, No, No, No, Yes, No]], [5, [Yes, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, No, No, No]], [2, [Yes, Yes, Yes, No, No, No]], [3, [Yes, No, No, Yes, No, No]], [4, [Yes, No, No, No, Yes, No]], [5, [Yes, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0001/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 2. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [3, [Yes, Yes, Yes, Yes, No, No, No, No]], [4, [Yes, No, Yes, No, Yes, No, Yes, No]], [5, [Yes, No, Yes, No, No, Yes, No, Yes]], [6, [Yes, No, Yes, No, Yes, No, Yes, No]], [7, [Yes, No, Yes, No, No, Yes, No, Yes]]], [0, 2], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [3, [Yes, Yes, Yes, Yes, No, No, No, No]], [4, [Yes, No, Yes, No, Yes, No, Yes, No]], [5, [Yes, No, Yes, No, No, Yes, No, Yes]], [6, [Yes, No, Yes, No, Yes, No, Yes, No]], [7, [Yes, No, Yes, No, No, Yes, No, Yes]]], [0, 2], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0002/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 3. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [5, [Yes, No, No, No, No, Yes, No, No, No, No]], [6, [Yes, No, No, No, No, No, Yes, No, No, No]], [7, [Yes, No, No, No, No, No, No, Yes, No, No]], [8, [Yes, No, No, No, No, No, No, No, Yes, No]], [9, [Yes, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [5, [Yes, No, No, No, No, Yes, No, No, No, No]], [6, [Yes, No, No, No, No, No, Yes, No, No, No]], [7, [Yes, No, No, No, No, No, No, Yes, No, No]], [8, [Yes, No, No, No, No, No, No, No, Yes, No]], [9, [Yes, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0003/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 4. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [6, [Yes, No, No, Yes, No, No, Yes, No, No, Yes, No, No]], [7, [Yes, No, No, Yes, No, No, No, Yes, No, No, Yes, No]], [8, [Yes, No, No, Yes, No, No, No, No, Yes, No, No, Yes]], [9, [Yes, No, No, Yes, No, No, Yes, No, No, Yes, No, No]], [10, [Yes, No, No, Yes, No, No, No, Yes, No, No, Yes, No]], [11, [Yes, No, No, Yes, No, No, No, No, Yes, No, No, Yes]]], [0, 3], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No]], [6, [Yes, No, No, Yes, No, No, Yes, No, No, Yes, No, No]], [7, [Yes, No, No, Yes, No, No, No, Yes, No, No, Yes, No]], [8, [Yes, No, No, Yes, No, No, No, No, Yes, No, No, Yes]], [9, [Yes, No, No, Yes, No, No, Yes, No, No, Yes, No, No]], [10, [Yes, No, No, Yes, No, No, No, Yes, No, No, Yes, No]], [11, [Yes, No, No, Yes, No, No, No, No, Yes, No, No, Yes]]], [0, 3], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0004/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 5. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [7, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [8, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [9, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [10, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [11, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [12, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No]], [7, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [8, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [9, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [10, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [11, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [12, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0005/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 6. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [8, [Yes, No, No, No, Yes, No, No, No, Yes, No, No, No, Yes, No, No, No]], [9, [Yes, No, No, No, Yes, No, No, No, No, Yes, No, No, No, Yes, No, No]], [10, [Yes, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, Yes, No]], [11, [Yes, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, Yes]], [12, [Yes, No, No, No, Yes, No, No, No, Yes, No, No, No, Yes, No, No, No]], [13, [Yes, No, No, No, Yes, No, No, No, No, Yes, No, No, No, Yes, No, No]], [14, [Yes, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, Yes, No]], [15, [Yes, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, Yes]]], [0, 4], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No]], [8, [Yes, No, No, No, Yes, No, No, No, Yes, No, No, No, Yes, No, No, No]], [9, [Yes, No, No, No, Yes, No, No, No, No, Yes, No, No, No, Yes, No, No]], [10, [Yes, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, Yes, No]], [11, [Yes, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, Yes]], [12, [Yes, No, No, No, Yes, No, No, No, Yes, No, No, No, Yes, No, No, No]], [13, [Yes, No, No, No, Yes, No, No, No, No, Yes, No, No, No, Yes, No, No]], [14, [Yes, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, Yes, No]], [15, [Yes, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, Yes]]], [0, 4], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0006/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 7. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [9, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [10, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [11, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [12, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No]], [9, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [10, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [11, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [12, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0007/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 8. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No]], [11, [Yes, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No]], [12, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No]], [13, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No]], [14, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes]], [15, [Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No]], [16, [Yes, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No]], [17, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No]], [18, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No]], [19, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes]]], [0, 5], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No]], [11, [Yes, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No]], [12, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No]], [13, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No]], [14, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes]], [15, [Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No, No]], [16, [Yes, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No, No]], [17, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No, No]], [18, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes, No]], [19, [Yes, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, Yes]]], [0, 5], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0008/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 9. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0009/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 10. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No]], [14, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No]], [15, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No]], [16, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No]], [17, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes]], [18, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No]], [20, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No]], [21, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No]], [22, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No]], [23, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes]]], [0, 6], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No]], [14, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No]], [15, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No]], [16, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No]], [17, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes]], [18, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No, No]], [20, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No, No]], [21, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No, No]], [22, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes, No]], [23, [Yes, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, Yes]]], [0, 6], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0010/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 11. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0011/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 12. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No]], [18, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No]], [19, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No]], [20, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes]], [21, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No]], [25, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No]], [26, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No]], [27, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes]]], [0, 7], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No]], [18, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No]], [19, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No]], [20, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes]], [21, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No, No]], [25, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No, No]], [26, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes, No]], [27, [Yes, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, Yes]]], [0, 7], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0012/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 13. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0013/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 14. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No]], [22, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No]], [23, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes]], [24, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No]], [30, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No]], [31, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes]]], [0, 8], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No]], [22, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No]], [23, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes]], [24, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No, No]], [30, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes, No]], [31, [Yes, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, Yes]]], [0, 8], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0014/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 15. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0015/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 16. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No]], [26, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes]], [27, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No]], [35, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes]]], [0, 9], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No]], [26, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes]], [27, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes, No]], [35, [Yes, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, Yes]]], [0, 9], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0016/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 17. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0017/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 18. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes]], [30, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes]]], [0, 10], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes]], [30, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, Yes]]], [0, 10], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0018/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 19. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0019/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 20. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes]]], [0, 11], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, Yes]]], [0, 11], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0020/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 21. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0021/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 22. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0, 12], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0, 12], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0022/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 23. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [48, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [49, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [48, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [49, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0023/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 24. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [48, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [49, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [50, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [51, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0, 13], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [48, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [49, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [50, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [51, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0, 13], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0024/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 25. The Center Requires Commutation with Every Group Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

View answer and explanation([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [48, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [49, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [50, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [51, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [52, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [53, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

For each z retain its entire commutation row against every g; the center consists exactly of all-true rows. A sample of g values or the centralizer of one chosen element is not the full center. The exact result, retaining the requested order, is ([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [13, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [14, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [15, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [16, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [17, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [18, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [19, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [20, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [21, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [22, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [23, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [24, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [25, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [26, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [27, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [28, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [29, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [30, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [31, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [32, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [33, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [34, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [35, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [36, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [37, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [38, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [39, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No]], [40, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No, No]], [41, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No, No]], [42, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No, No]], [43, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [44, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [45, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [46, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [47, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [48, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [49, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [50, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [51, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [52, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [53, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0025/)

Center of a finite group · qf-mod46-center-commutation-table · candidate





## 26. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0026/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 27. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 2], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 2], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0027/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 28. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0028/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 29. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 3], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 3], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0029/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 30. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0030/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 31. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 4], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 4], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0031/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 32. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0032/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 33. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 5], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 5], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0033/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 34. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0034/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 35. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 6], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 6], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0035/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 36. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0036/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 37. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 7], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 7], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0037/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 38. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0038/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 39. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 8], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 8], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0039/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 40. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0040/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 41. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 9], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 9], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0041/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 42. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0042/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 43. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 10], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 10], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0043/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 44. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0044/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 45. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 11], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 11], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0045/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 46. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0046/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 47. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 12], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 12], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0047/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 48. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0048/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 49. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 13], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0, 13], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0049/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 50. Identity, Product and Inverse Closure Certify the Center as a Subgroup

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

Compute the whole center first, then check identity membership, every center product and every center inverse separately. Do not replace these obligations with a sampled commutation check or invoke a class equation. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0050/)

Center of a finite group · qf-mod46-center-subgroup-audit · candidate





## 51. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0051/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 52. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 2], [[0, [0, 2], Yes], [1, [0, 2], Yes], [2, [0, 2], Yes], [3, [0, 2], Yes], [4, [0, 2], Yes], [5, [0, 2], Yes], [6, [0, 2], Yes], [7, [0, 2], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 2], [[0, [0, 2], Yes], [1, [0, 2], Yes], [2, [0, 2], Yes], [3, [0, 2], Yes], [4, [0, 2], Yes], [5, [0, 2], Yes], [6, [0, 2], Yes], [7, [0, 2], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0052/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 53. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0053/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 54. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 3], [[0, [0, 3], Yes], [1, [0, 3], Yes], [2, [0, 3], Yes], [3, [0, 3], Yes], [4, [0, 3], Yes], [5, [0, 3], Yes], [6, [0, 3], Yes], [7, [0, 3], Yes], [8, [0, 3], Yes], [9, [0, 3], Yes], [10, [0, 3], Yes], [11, [0, 3], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 3], [[0, [0, 3], Yes], [1, [0, 3], Yes], [2, [0, 3], Yes], [3, [0, 3], Yes], [4, [0, 3], Yes], [5, [0, 3], Yes], [6, [0, 3], Yes], [7, [0, 3], Yes], [8, [0, 3], Yes], [9, [0, 3], Yes], [10, [0, 3], Yes], [11, [0, 3], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0054/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 55. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0055/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 56. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 4], [[0, [0, 4], Yes], [1, [0, 4], Yes], [2, [0, 4], Yes], [3, [0, 4], Yes], [4, [0, 4], Yes], [5, [0, 4], Yes], [6, [0, 4], Yes], [7, [0, 4], Yes], [8, [0, 4], Yes], [9, [0, 4], Yes], [10, [0, 4], Yes], [11, [0, 4], Yes], [12, [0, 4], Yes], [13, [0, 4], Yes], [14, [0, 4], Yes], [15, [0, 4], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 4], [[0, [0, 4], Yes], [1, [0, 4], Yes], [2, [0, 4], Yes], [3, [0, 4], Yes], [4, [0, 4], Yes], [5, [0, 4], Yes], [6, [0, 4], Yes], [7, [0, 4], Yes], [8, [0, 4], Yes], [9, [0, 4], Yes], [10, [0, 4], Yes], [11, [0, 4], Yes], [12, [0, 4], Yes], [13, [0, 4], Yes], [14, [0, 4], Yes], [15, [0, 4], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0056/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 57. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0057/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 58. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 5], [[0, [0, 5], Yes], [1, [0, 5], Yes], [2, [0, 5], Yes], [3, [0, 5], Yes], [4, [0, 5], Yes], [5, [0, 5], Yes], [6, [0, 5], Yes], [7, [0, 5], Yes], [8, [0, 5], Yes], [9, [0, 5], Yes], [10, [0, 5], Yes], [11, [0, 5], Yes], [12, [0, 5], Yes], [13, [0, 5], Yes], [14, [0, 5], Yes], [15, [0, 5], Yes], [16, [0, 5], Yes], [17, [0, 5], Yes], [18, [0, 5], Yes], [19, [0, 5], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 5], [[0, [0, 5], Yes], [1, [0, 5], Yes], [2, [0, 5], Yes], [3, [0, 5], Yes], [4, [0, 5], Yes], [5, [0, 5], Yes], [6, [0, 5], Yes], [7, [0, 5], Yes], [8, [0, 5], Yes], [9, [0, 5], Yes], [10, [0, 5], Yes], [11, [0, 5], Yes], [12, [0, 5], Yes], [13, [0, 5], Yes], [14, [0, 5], Yes], [15, [0, 5], Yes], [16, [0, 5], Yes], [17, [0, 5], Yes], [18, [0, 5], Yes], [19, [0, 5], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0058/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 59. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0059/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 60. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 6], [[0, [0, 6], Yes], [1, [0, 6], Yes], [2, [0, 6], Yes], [3, [0, 6], Yes], [4, [0, 6], Yes], [5, [0, 6], Yes], [6, [0, 6], Yes], [7, [0, 6], Yes], [8, [0, 6], Yes], [9, [0, 6], Yes], [10, [0, 6], Yes], [11, [0, 6], Yes], [12, [0, 6], Yes], [13, [0, 6], Yes], [14, [0, 6], Yes], [15, [0, 6], Yes], [16, [0, 6], Yes], [17, [0, 6], Yes], [18, [0, 6], Yes], [19, [0, 6], Yes], [20, [0, 6], Yes], [21, [0, 6], Yes], [22, [0, 6], Yes], [23, [0, 6], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 6], [[0, [0, 6], Yes], [1, [0, 6], Yes], [2, [0, 6], Yes], [3, [0, 6], Yes], [4, [0, 6], Yes], [5, [0, 6], Yes], [6, [0, 6], Yes], [7, [0, 6], Yes], [8, [0, 6], Yes], [9, [0, 6], Yes], [10, [0, 6], Yes], [11, [0, 6], Yes], [12, [0, 6], Yes], [13, [0, 6], Yes], [14, [0, 6], Yes], [15, [0, 6], Yes], [16, [0, 6], Yes], [17, [0, 6], Yes], [18, [0, 6], Yes], [19, [0, 6], Yes], [20, [0, 6], Yes], [21, [0, 6], Yes], [22, [0, 6], Yes], [23, [0, 6], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0060/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 61. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0061/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 62. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 7], [[0, [0, 7], Yes], [1, [0, 7], Yes], [2, [0, 7], Yes], [3, [0, 7], Yes], [4, [0, 7], Yes], [5, [0, 7], Yes], [6, [0, 7], Yes], [7, [0, 7], Yes], [8, [0, 7], Yes], [9, [0, 7], Yes], [10, [0, 7], Yes], [11, [0, 7], Yes], [12, [0, 7], Yes], [13, [0, 7], Yes], [14, [0, 7], Yes], [15, [0, 7], Yes], [16, [0, 7], Yes], [17, [0, 7], Yes], [18, [0, 7], Yes], [19, [0, 7], Yes], [20, [0, 7], Yes], [21, [0, 7], Yes], [22, [0, 7], Yes], [23, [0, 7], Yes], [24, [0, 7], Yes], [25, [0, 7], Yes], [26, [0, 7], Yes], [27, [0, 7], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 7], [[0, [0, 7], Yes], [1, [0, 7], Yes], [2, [0, 7], Yes], [3, [0, 7], Yes], [4, [0, 7], Yes], [5, [0, 7], Yes], [6, [0, 7], Yes], [7, [0, 7], Yes], [8, [0, 7], Yes], [9, [0, 7], Yes], [10, [0, 7], Yes], [11, [0, 7], Yes], [12, [0, 7], Yes], [13, [0, 7], Yes], [14, [0, 7], Yes], [15, [0, 7], Yes], [16, [0, 7], Yes], [17, [0, 7], Yes], [18, [0, 7], Yes], [19, [0, 7], Yes], [20, [0, 7], Yes], [21, [0, 7], Yes], [22, [0, 7], Yes], [23, [0, 7], Yes], [24, [0, 7], Yes], [25, [0, 7], Yes], [26, [0, 7], Yes], [27, [0, 7], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0062/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 63. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0063/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 64. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 8], [[0, [0, 8], Yes], [1, [0, 8], Yes], [2, [0, 8], Yes], [3, [0, 8], Yes], [4, [0, 8], Yes], [5, [0, 8], Yes], [6, [0, 8], Yes], [7, [0, 8], Yes], [8, [0, 8], Yes], [9, [0, 8], Yes], [10, [0, 8], Yes], [11, [0, 8], Yes], [12, [0, 8], Yes], [13, [0, 8], Yes], [14, [0, 8], Yes], [15, [0, 8], Yes], [16, [0, 8], Yes], [17, [0, 8], Yes], [18, [0, 8], Yes], [19, [0, 8], Yes], [20, [0, 8], Yes], [21, [0, 8], Yes], [22, [0, 8], Yes], [23, [0, 8], Yes], [24, [0, 8], Yes], [25, [0, 8], Yes], [26, [0, 8], Yes], [27, [0, 8], Yes], [28, [0, 8], Yes], [29, [0, 8], Yes], [30, [0, 8], Yes], [31, [0, 8], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 8], [[0, [0, 8], Yes], [1, [0, 8], Yes], [2, [0, 8], Yes], [3, [0, 8], Yes], [4, [0, 8], Yes], [5, [0, 8], Yes], [6, [0, 8], Yes], [7, [0, 8], Yes], [8, [0, 8], Yes], [9, [0, 8], Yes], [10, [0, 8], Yes], [11, [0, 8], Yes], [12, [0, 8], Yes], [13, [0, 8], Yes], [14, [0, 8], Yes], [15, [0, 8], Yes], [16, [0, 8], Yes], [17, [0, 8], Yes], [18, [0, 8], Yes], [19, [0, 8], Yes], [20, [0, 8], Yes], [21, [0, 8], Yes], [22, [0, 8], Yes], [23, [0, 8], Yes], [24, [0, 8], Yes], [25, [0, 8], Yes], [26, [0, 8], Yes], [27, [0, 8], Yes], [28, [0, 8], Yes], [29, [0, 8], Yes], [30, [0, 8], Yes], [31, [0, 8], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0064/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 65. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0065/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 66. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 9], [[0, [0, 9], Yes], [1, [0, 9], Yes], [2, [0, 9], Yes], [3, [0, 9], Yes], [4, [0, 9], Yes], [5, [0, 9], Yes], [6, [0, 9], Yes], [7, [0, 9], Yes], [8, [0, 9], Yes], [9, [0, 9], Yes], [10, [0, 9], Yes], [11, [0, 9], Yes], [12, [0, 9], Yes], [13, [0, 9], Yes], [14, [0, 9], Yes], [15, [0, 9], Yes], [16, [0, 9], Yes], [17, [0, 9], Yes], [18, [0, 9], Yes], [19, [0, 9], Yes], [20, [0, 9], Yes], [21, [0, 9], Yes], [22, [0, 9], Yes], [23, [0, 9], Yes], [24, [0, 9], Yes], [25, [0, 9], Yes], [26, [0, 9], Yes], [27, [0, 9], Yes], [28, [0, 9], Yes], [29, [0, 9], Yes], [30, [0, 9], Yes], [31, [0, 9], Yes], [32, [0, 9], Yes], [33, [0, 9], Yes], [34, [0, 9], Yes], [35, [0, 9], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 9], [[0, [0, 9], Yes], [1, [0, 9], Yes], [2, [0, 9], Yes], [3, [0, 9], Yes], [4, [0, 9], Yes], [5, [0, 9], Yes], [6, [0, 9], Yes], [7, [0, 9], Yes], [8, [0, 9], Yes], [9, [0, 9], Yes], [10, [0, 9], Yes], [11, [0, 9], Yes], [12, [0, 9], Yes], [13, [0, 9], Yes], [14, [0, 9], Yes], [15, [0, 9], Yes], [16, [0, 9], Yes], [17, [0, 9], Yes], [18, [0, 9], Yes], [19, [0, 9], Yes], [20, [0, 9], Yes], [21, [0, 9], Yes], [22, [0, 9], Yes], [23, [0, 9], Yes], [24, [0, 9], Yes], [25, [0, 9], Yes], [26, [0, 9], Yes], [27, [0, 9], Yes], [28, [0, 9], Yes], [29, [0, 9], Yes], [30, [0, 9], Yes], [31, [0, 9], Yes], [32, [0, 9], Yes], [33, [0, 9], Yes], [34, [0, 9], Yes], [35, [0, 9], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0066/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 67. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0067/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 68. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 10], [[0, [0, 10], Yes], [1, [0, 10], Yes], [2, [0, 10], Yes], [3, [0, 10], Yes], [4, [0, 10], Yes], [5, [0, 10], Yes], [6, [0, 10], Yes], [7, [0, 10], Yes], [8, [0, 10], Yes], [9, [0, 10], Yes], [10, [0, 10], Yes], [11, [0, 10], Yes], [12, [0, 10], Yes], [13, [0, 10], Yes], [14, [0, 10], Yes], [15, [0, 10], Yes], [16, [0, 10], Yes], [17, [0, 10], Yes], [18, [0, 10], Yes], [19, [0, 10], Yes], [20, [0, 10], Yes], [21, [0, 10], Yes], [22, [0, 10], Yes], [23, [0, 10], Yes], [24, [0, 10], Yes], [25, [0, 10], Yes], [26, [0, 10], Yes], [27, [0, 10], Yes], [28, [0, 10], Yes], [29, [0, 10], Yes], [30, [0, 10], Yes], [31, [0, 10], Yes], [32, [0, 10], Yes], [33, [0, 10], Yes], [34, [0, 10], Yes], [35, [0, 10], Yes], [36, [0, 10], Yes], [37, [0, 10], Yes], [38, [0, 10], Yes], [39, [0, 10], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 10], [[0, [0, 10], Yes], [1, [0, 10], Yes], [2, [0, 10], Yes], [3, [0, 10], Yes], [4, [0, 10], Yes], [5, [0, 10], Yes], [6, [0, 10], Yes], [7, [0, 10], Yes], [8, [0, 10], Yes], [9, [0, 10], Yes], [10, [0, 10], Yes], [11, [0, 10], Yes], [12, [0, 10], Yes], [13, [0, 10], Yes], [14, [0, 10], Yes], [15, [0, 10], Yes], [16, [0, 10], Yes], [17, [0, 10], Yes], [18, [0, 10], Yes], [19, [0, 10], Yes], [20, [0, 10], Yes], [21, [0, 10], Yes], [22, [0, 10], Yes], [23, [0, 10], Yes], [24, [0, 10], Yes], [25, [0, 10], Yes], [26, [0, 10], Yes], [27, [0, 10], Yes], [28, [0, 10], Yes], [29, [0, 10], Yes], [30, [0, 10], Yes], [31, [0, 10], Yes], [32, [0, 10], Yes], [33, [0, 10], Yes], [34, [0, 10], Yes], [35, [0, 10], Yes], [36, [0, 10], Yes], [37, [0, 10], Yes], [38, [0, 10], Yes], [39, [0, 10], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0068/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 69. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0069/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 70. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 11], [[0, [0, 11], Yes], [1, [0, 11], Yes], [2, [0, 11], Yes], [3, [0, 11], Yes], [4, [0, 11], Yes], [5, [0, 11], Yes], [6, [0, 11], Yes], [7, [0, 11], Yes], [8, [0, 11], Yes], [9, [0, 11], Yes], [10, [0, 11], Yes], [11, [0, 11], Yes], [12, [0, 11], Yes], [13, [0, 11], Yes], [14, [0, 11], Yes], [15, [0, 11], Yes], [16, [0, 11], Yes], [17, [0, 11], Yes], [18, [0, 11], Yes], [19, [0, 11], Yes], [20, [0, 11], Yes], [21, [0, 11], Yes], [22, [0, 11], Yes], [23, [0, 11], Yes], [24, [0, 11], Yes], [25, [0, 11], Yes], [26, [0, 11], Yes], [27, [0, 11], Yes], [28, [0, 11], Yes], [29, [0, 11], Yes], [30, [0, 11], Yes], [31, [0, 11], Yes], [32, [0, 11], Yes], [33, [0, 11], Yes], [34, [0, 11], Yes], [35, [0, 11], Yes], [36, [0, 11], Yes], [37, [0, 11], Yes], [38, [0, 11], Yes], [39, [0, 11], Yes], [40, [0, 11], Yes], [41, [0, 11], Yes], [42, [0, 11], Yes], [43, [0, 11], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 11], [[0, [0, 11], Yes], [1, [0, 11], Yes], [2, [0, 11], Yes], [3, [0, 11], Yes], [4, [0, 11], Yes], [5, [0, 11], Yes], [6, [0, 11], Yes], [7, [0, 11], Yes], [8, [0, 11], Yes], [9, [0, 11], Yes], [10, [0, 11], Yes], [11, [0, 11], Yes], [12, [0, 11], Yes], [13, [0, 11], Yes], [14, [0, 11], Yes], [15, [0, 11], Yes], [16, [0, 11], Yes], [17, [0, 11], Yes], [18, [0, 11], Yes], [19, [0, 11], Yes], [20, [0, 11], Yes], [21, [0, 11], Yes], [22, [0, 11], Yes], [23, [0, 11], Yes], [24, [0, 11], Yes], [25, [0, 11], Yes], [26, [0, 11], Yes], [27, [0, 11], Yes], [28, [0, 11], Yes], [29, [0, 11], Yes], [30, [0, 11], Yes], [31, [0, 11], Yes], [32, [0, 11], Yes], [33, [0, 11], Yes], [34, [0, 11], Yes], [35, [0, 11], Yes], [36, [0, 11], Yes], [37, [0, 11], Yes], [38, [0, 11], Yes], [39, [0, 11], Yes], [40, [0, 11], Yes], [41, [0, 11], Yes], [42, [0, 11], Yes], [43, [0, 11], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0070/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 71. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes], [42, [0], Yes], [43, [0], Yes], [44, [0], Yes], [45, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes], [42, [0], Yes], [43, [0], Yes], [44, [0], Yes], [45, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0071/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 72. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 12], [[0, [0, 12], Yes], [1, [0, 12], Yes], [2, [0, 12], Yes], [3, [0, 12], Yes], [4, [0, 12], Yes], [5, [0, 12], Yes], [6, [0, 12], Yes], [7, [0, 12], Yes], [8, [0, 12], Yes], [9, [0, 12], Yes], [10, [0, 12], Yes], [11, [0, 12], Yes], [12, [0, 12], Yes], [13, [0, 12], Yes], [14, [0, 12], Yes], [15, [0, 12], Yes], [16, [0, 12], Yes], [17, [0, 12], Yes], [18, [0, 12], Yes], [19, [0, 12], Yes], [20, [0, 12], Yes], [21, [0, 12], Yes], [22, [0, 12], Yes], [23, [0, 12], Yes], [24, [0, 12], Yes], [25, [0, 12], Yes], [26, [0, 12], Yes], [27, [0, 12], Yes], [28, [0, 12], Yes], [29, [0, 12], Yes], [30, [0, 12], Yes], [31, [0, 12], Yes], [32, [0, 12], Yes], [33, [0, 12], Yes], [34, [0, 12], Yes], [35, [0, 12], Yes], [36, [0, 12], Yes], [37, [0, 12], Yes], [38, [0, 12], Yes], [39, [0, 12], Yes], [40, [0, 12], Yes], [41, [0, 12], Yes], [42, [0, 12], Yes], [43, [0, 12], Yes], [44, [0, 12], Yes], [45, [0, 12], Yes], [46, [0, 12], Yes], [47, [0, 12], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 12], [[0, [0, 12], Yes], [1, [0, 12], Yes], [2, [0, 12], Yes], [3, [0, 12], Yes], [4, [0, 12], Yes], [5, [0, 12], Yes], [6, [0, 12], Yes], [7, [0, 12], Yes], [8, [0, 12], Yes], [9, [0, 12], Yes], [10, [0, 12], Yes], [11, [0, 12], Yes], [12, [0, 12], Yes], [13, [0, 12], Yes], [14, [0, 12], Yes], [15, [0, 12], Yes], [16, [0, 12], Yes], [17, [0, 12], Yes], [18, [0, 12], Yes], [19, [0, 12], Yes], [20, [0, 12], Yes], [21, [0, 12], Yes], [22, [0, 12], Yes], [23, [0, 12], Yes], [24, [0, 12], Yes], [25, [0, 12], Yes], [26, [0, 12], Yes], [27, [0, 12], Yes], [28, [0, 12], Yes], [29, [0, 12], Yes], [30, [0, 12], Yes], [31, [0, 12], Yes], [32, [0, 12], Yes], [33, [0, 12], Yes], [34, [0, 12], Yes], [35, [0, 12], Yes], [36, [0, 12], Yes], [37, [0, 12], Yes], [38, [0, 12], Yes], [39, [0, 12], Yes], [40, [0, 12], Yes], [41, [0, 12], Yes], [42, [0, 12], Yes], [43, [0, 12], Yes], [44, [0, 12], Yes], [45, [0, 12], Yes], [46, [0, 12], Yes], [47, [0, 12], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0072/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 73. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes], [42, [0], Yes], [43, [0], Yes], [44, [0], Yes], [45, [0], Yes], [46, [0], Yes], [47, [0], Yes], [48, [0], Yes], [49, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes], [42, [0], Yes], [43, [0], Yes], [44, [0], Yes], [45, [0], Yes], [46, [0], Yes], [47, [0], Yes], [48, [0], Yes], [49, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0073/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 74. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0, 13], [[0, [0, 13], Yes], [1, [0, 13], Yes], [2, [0, 13], Yes], [3, [0, 13], Yes], [4, [0, 13], Yes], [5, [0, 13], Yes], [6, [0, 13], Yes], [7, [0, 13], Yes], [8, [0, 13], Yes], [9, [0, 13], Yes], [10, [0, 13], Yes], [11, [0, 13], Yes], [12, [0, 13], Yes], [13, [0, 13], Yes], [14, [0, 13], Yes], [15, [0, 13], Yes], [16, [0, 13], Yes], [17, [0, 13], Yes], [18, [0, 13], Yes], [19, [0, 13], Yes], [20, [0, 13], Yes], [21, [0, 13], Yes], [22, [0, 13], Yes], [23, [0, 13], Yes], [24, [0, 13], Yes], [25, [0, 13], Yes], [26, [0, 13], Yes], [27, [0, 13], Yes], [28, [0, 13], Yes], [29, [0, 13], Yes], [30, [0, 13], Yes], [31, [0, 13], Yes], [32, [0, 13], Yes], [33, [0, 13], Yes], [34, [0, 13], Yes], [35, [0, 13], Yes], [36, [0, 13], Yes], [37, [0, 13], Yes], [38, [0, 13], Yes], [39, [0, 13], Yes], [40, [0, 13], Yes], [41, [0, 13], Yes], [42, [0, 13], Yes], [43, [0, 13], Yes], [44, [0, 13], Yes], [45, [0, 13], Yes], [46, [0, 13], Yes], [47, [0, 13], Yes], [48, [0, 13], Yes], [49, [0, 13], Yes], [50, [0, 13], Yes], [51, [0, 13], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0, 13], [[0, [0, 13], Yes], [1, [0, 13], Yes], [2, [0, 13], Yes], [3, [0, 13], Yes], [4, [0, 13], Yes], [5, [0, 13], Yes], [6, [0, 13], Yes], [7, [0, 13], Yes], [8, [0, 13], Yes], [9, [0, 13], Yes], [10, [0, 13], Yes], [11, [0, 13], Yes], [12, [0, 13], Yes], [13, [0, 13], Yes], [14, [0, 13], Yes], [15, [0, 13], Yes], [16, [0, 13], Yes], [17, [0, 13], Yes], [18, [0, 13], Yes], [19, [0, 13], Yes], [20, [0, 13], Yes], [21, [0, 13], Yes], [22, [0, 13], Yes], [23, [0, 13], Yes], [24, [0, 13], Yes], [25, [0, 13], Yes], [26, [0, 13], Yes], [27, [0, 13], Yes], [28, [0, 13], Yes], [29, [0, 13], Yes], [30, [0, 13], Yes], [31, [0, 13], Yes], [32, [0, 13], Yes], [33, [0, 13], Yes], [34, [0, 13], Yes], [35, [0, 13], Yes], [36, [0, 13], Yes], [37, [0, 13], Yes], [38, [0, 13], Yes], [39, [0, 13], Yes], [40, [0, 13], Yes], [41, [0, 13], Yes], [42, [0, 13], Yes], [43, [0, 13], Yes], [44, [0, 13], Yes], [45, [0, 13], Yes], [46, [0, 13], Yes], [47, [0, 13], Yes], [48, [0, 13], Yes], [49, [0, 13], Yes], [50, [0, 13], Yes], [51, [0, 13], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0074/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 75. Every Group Conjugation Preserves Each Central Element

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, conjugate_rows, normal, certificate).

View answer and explanation([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes], [42, [0], Yes], [43, [0], Yes], [44, [0], Yes], [45, [0], Yes], [46, [0], Yes], [47, [0], Yes], [48, [0], Yes], [49, [0], Yes], [50, [0], Yes], [51, [0], Yes], [52, [0], Yes], [53, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element)

Retain all conjugate-center rows and normality. For central z, g*z*g^(-1)=z for every g. This is the complete normal-subgroup audit, not a claim of characteristic status or a class-equation result. The exact result, retaining the requested order, is ([0], [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes], [24, [0], Yes], [25, [0], Yes], [26, [0], Yes], [27, [0], Yes], [28, [0], Yes], [29, [0], Yes], [30, [0], Yes], [31, [0], Yes], [32, [0], Yes], [33, [0], Yes], [34, [0], Yes], [35, [0], Yes], [36, [0], Yes], [37, [0], Yes], [38, [0], Yes], [39, [0], Yes], [40, [0], Yes], [41, [0], Yes], [42, [0], Yes], [43, [0], Yes], [44, [0], Yes], [45, [0], Yes], [46, [0], Yes], [47, [0], Yes], [48, [0], Yes], [49, [0], Yes], [50, [0], Yes], [51, [0], Yes], [52, [0], Yes], [53, [0], Yes]], Yes, every_conjugate_of_a_central_element_remains_the_same_central_element).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0075/)

Center of a finite group · qf-mod46-center-normality-audit · candidate





## 76. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=3. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(3, No, [3, 4, 5], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (3, No, [3, 4, 5], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0076/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 77. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=4. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(4, No, [4, 6], No, Yes, [0, 2], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (4, No, [4, 6], No, Yes, [0, 2], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0077/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 78. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=5. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(5, No, [5, 6, 7, 8, 9], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (5, No, [5, 6, 7, 8, 9], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0078/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 79. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=3. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(3, Yes, [3], Yes, Yes, [0, 3], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (3, Yes, [3], Yes, Yes, [0, 3], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0079/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 80. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=7. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(7, No, [7, 8, 9, 10, 11, 12, 13], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (7, No, [7, 8, 9, 10, 11, 12, 13], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0080/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 81. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=8. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(8, No, [8, 10, 12, 14], No, Yes, [0, 4], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (8, No, [8, 10, 12, 14], No, Yes, [0, 4], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0081/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 82. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=9. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(9, No, [9, 10, 11, 12, 13, 14, 15, 16, 17], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (9, No, [9, 10, 11, 12, 13, 14, 15, 16, 17], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0082/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 83. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=10. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(10, No, [10, 12, 14, 16, 18], No, Yes, [0, 5], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (10, No, [10, 12, 14, 16, 18], No, Yes, [0, 5], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0083/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 84. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=11. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(11, No, [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (11, No, [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0084/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 85. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=6. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(6, Yes, [6], Yes, Yes, [0, 6], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (6, Yes, [6], Yes, Yes, [0, 6], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0085/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 86. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=13. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(13, No, [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (13, No, [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0086/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 87. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=14. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(14, No, [14, 16, 18, 20, 22, 24, 26], No, Yes, [0, 7], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (14, No, [14, 16, 18, 20, 22, 24, 26], No, Yes, [0, 7], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0087/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 88. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=15. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(15, No, [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (15, No, [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0088/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 89. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=16. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(16, No, [16, 18, 20, 22, 24, 26, 28, 30], No, Yes, [0, 8], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (16, No, [16, 18, 20, 22, 24, 26, 28, 30], No, Yes, [0, 8], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0089/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 90. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=17. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(17, No, [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (17, No, [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0090/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 91. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=9. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(9, Yes, [9], Yes, Yes, [0, 9], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (9, Yes, [9], Yes, Yes, [0, 9], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0091/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 92. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=19. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(19, No, [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (19, No, [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0092/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 93. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=20. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(20, No, [20, 22, 24, 26, 28, 30, 32, 34, 36, 38], No, Yes, [0, 10], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (20, No, [20, 22, 24, 26, 28, 30, 32, 34, 36, 38], No, Yes, [0, 10], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0093/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 94. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=21. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(21, No, [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (21, No, [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0094/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 95. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=22. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(22, No, [22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], No, Yes, [0, 11], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (22, No, [22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], No, Yes, [0, 11], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0095/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 96. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=23. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(23, No, [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (23, No, [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0096/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 97. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=12. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(12, Yes, [12], Yes, Yes, [0, 12], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (12, Yes, [12], Yes, Yes, [0, 12], No, No, Yes, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0097/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 98. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=25. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(25, No, [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (25, No, [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0098/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 99. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=26. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(26, No, [26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], No, Yes, [0, 13], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (26, No, [26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], No, Yes, [0, 13], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0099/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate





## 100. Centrality, Singleton Conjugacy and Whole-Group Abelianness Have Separate Flags

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=27. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).

Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer.

View answer and explanation(27, No, [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)

Report centrality and its full conjugacy class, comparing centrality with singleton status. Separately compare abelianness with center=G. Both conditions may be No while an equivalence flag is Yes. A noncentral element gets the first noncommuting g; otherwise the witness is null. The exact result, retaining the requested order, is (27, No, [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], No, Yes, [0], No, No, Yes, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r02-finite-group-center-0100/)

Center of a finite group · qf-mod46-center-abelian-conjugacy-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r02-finite-group-center-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261115-r02-finite-group-center-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r02-finite-group-center-foundations/validation-report.json)