# Qualified Finite Groups, Commutativity, Orders and Cyclic Subgroups

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-20261013-advance-20261026-r01-finite-abelian-group-foundations · 2026-10-13 · 100 questions

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



## 1. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0001/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 2. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0002/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 3. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0003/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 4. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0004/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 5. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0005/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 6. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0006/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 7. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0007/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 8. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0008/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 9. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0009/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 10. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0010/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 11. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0011/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 12. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0012/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 13. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0013/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 14. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0014/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 15. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0015/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 16. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0016/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 17. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0017/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 18. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0018/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 19. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0019/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 20. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0020/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 21. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0021/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 22. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0022/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 23. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0023/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 24. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0024/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 25. Audit Every Pair of a Qualified Finite Group

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).

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

Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0025/)

Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate





## 26. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(1, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0026/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 27. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(0, 0, 1, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0027/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 28. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(5, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0028/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 29. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(4, 0, 2, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0029/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 30. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(3, 0, 3, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0030/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 31. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(2, 0, 3, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0031/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 32. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(1, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0032/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 33. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(0, 0, 1, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0033/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 34. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(5, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0034/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 35. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(4, 0, 2, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0035/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 36. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(3, 0, 3, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0036/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 37. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(2, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0037/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 38. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(1, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0038/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 39. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(0, 0, 1, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0039/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 40. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(5, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0040/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 41. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(4, 0, 3, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0041/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 42. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(3, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0042/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 43. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(2, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0043/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 44. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(1, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0044/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 45. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(0, 0, 1, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0045/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 46. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(5, 0, 3, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0046/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 47. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(4, 0, 3, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0047/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 48. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(3, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0048/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 49. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(2, 0, 6, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0049/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 50. Element Order Starts from the Actual Identity

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).

View answer and explanation(1, 0, 3, Yes)

Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0050/)

Finite abelian group · qf-mod26-abelian-element-order · candidate





## 51. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=3. Give (element,multiplier,identity,result).

View answer and explanation(1, 3, 0, 3)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0051/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 52. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=4. Give (element,multiplier,identity,result).

View answer and explanation(0, 4, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 4, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0052/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 53. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=5. Give (element,multiplier,identity,result).

View answer and explanation(5, 5, 0, 1)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 5, 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0053/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 54. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=6. Give (element,multiplier,identity,result).

View answer and explanation(4, 6, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0054/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 55. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=7. Give (element,multiplier,identity,result).

View answer and explanation(3, 7, 0, 3)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 7, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0055/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 56. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=8. Give (element,multiplier,identity,result).

View answer and explanation(2, 8, 0, 4)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 8, 0, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0056/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 57. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=9. Give (element,multiplier,identity,result).

View answer and explanation(1, 9, 0, 2)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 9, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0057/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 58. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=10. Give (element,multiplier,identity,result).

View answer and explanation(0, 10, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 10, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0058/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 59. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=11. Give (element,multiplier,identity,result).

View answer and explanation(5, 11, 0, 1)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 11, 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0059/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 60. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=12. Give (element,multiplier,identity,result).

View answer and explanation(4, 12, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 12, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0060/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 61. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=13. Give (element,multiplier,identity,result).

View answer and explanation(3, 13, 0, 3)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 13, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0061/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 62. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=14. Give (element,multiplier,identity,result).

View answer and explanation(2, 14, 0, 4)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 14, 0, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0062/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 63. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=15. Give (element,multiplier,identity,result).

View answer and explanation(1, 15, 0, 2)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 15, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0063/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 64. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=16. Give (element,multiplier,identity,result).

View answer and explanation(0, 16, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 16, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0064/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 65. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=17. Give (element,multiplier,identity,result).

View answer and explanation(5, 17, 0, 1)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 17, 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0065/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 66. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=18. Give (element,multiplier,identity,result).

View answer and explanation(4, 18, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 18, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0066/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 67. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=19. Give (element,multiplier,identity,result).

View answer and explanation(3, 19, 0, 3)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 19, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0067/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 68. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=20. Give (element,multiplier,identity,result).

View answer and explanation(2, 20, 0, 3)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 20, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0068/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 69. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=21. Give (element,multiplier,identity,result).

View answer and explanation(1, 21, 0, 2)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 21, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0069/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 70. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=22. Give (element,multiplier,identity,result).

View answer and explanation(0, 22, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 22, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0070/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 71. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=23. Give (element,multiplier,identity,result).

View answer and explanation(5, 23, 0, 2)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 23, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0071/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 72. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=24. Give (element,multiplier,identity,result).

View answer and explanation(4, 24, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 24, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0072/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 73. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=25. Give (element,multiplier,identity,result).

View answer and explanation(3, 25, 0, 3)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 25, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0073/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 74. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=26. Give (element,multiplier,identity,result).

View answer and explanation(2, 26, 0, 3)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 26, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0074/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 75. Repeated Group Operation with a Zero-Multiple Boundary

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=27. Give (element,multiplier,identity,result).

View answer and explanation(1, 27, 0, 0)

Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 27, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0075/)

Finite abelian group · qf-mod26-abelian-integer-multiple · candidate





## 76. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 2, 3, 4, 5], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0076/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 77. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(0, 0, [0], 0, 1)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0077/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 78. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 3, 4, 2, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0078/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 79. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(4, 0, [0, 4], 0, 2)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4], 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0079/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 80. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(3, 0, [0, 3, 2], 0, 3)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2], 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0080/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 81. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(2, 0, [0, 2, 4], 0, 3)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 4], 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0081/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 82. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 3, 2, 4, 5], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0082/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 83. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(0, 0, [0], 0, 1)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0083/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 84. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 2, 4, 3, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0084/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 85. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(4, 0, [0, 4], 0, 2)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4], 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0085/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 86. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(3, 0, [0, 3, 2], 0, 3)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2], 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0086/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 87. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 4, 5, 3, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0087/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 88. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 4, 2, 3, 5], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0088/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 89. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(0, 0, [0], 0, 1)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0089/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 90. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 2, 3, 4, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0090/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 91. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(4, 0, [0, 4, 5], 0, 3)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4, 5], 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0091/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 92. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2, 5, 4, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0092/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 93. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 3, 5, 4, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0093/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 94. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 5, 2, 3, 4], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0094/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 95. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(0, 0, [0], 0, 1)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0095/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 96. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(5, 0, [0, 5, 2], 0, 3)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 2], 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0096/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 97. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(4, 0, [0, 4, 5], 0, 3)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4, 5], 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0097/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 98. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2, 4, 5, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0098/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 99. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).

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

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 3, 4, 5, 1], 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0099/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate





## 100. Preserve the Generation Order of One Cyclic Subgroup

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).

View answer and explanation(1, 0, [0, 1, 4], 0, 3)

Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 4], 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0100/)

Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r01-finite-abelian-group-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261026-r01-finite-abelian-group-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r01-finite-abelian-group-foundations/validation-report.json)