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Question 25678: Identity, Product and Inverse Closure Certify the Center as a Subgroup

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0050-25678 · q-20261015-advance-20261115-r02-finite-group-center-0028

Original question and givens

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

  • G001 — Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (center, contains_identity, product_closed, inverse_closed, subgroup, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [center-subgroup-audit]

After

derived = [[center, contains_identity, product_closed, inverse_closed, subgroup, certificate]]

Conditions: none

List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [Center ORIGINAL group, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0]

After

derived = [10, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9]], Yes]

Conditions: none

Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0.

Step identifiers and dependencies

Rule: Group_complete_table_actual_identity_bound64 · Uses: G001, S001

S003 · implies

Before

source roles = [Center ORIGINAL group, 10, 0]

After

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 1000, [0, 4, 3, 2, 1, 5, 6, 7, 8, 9]]

Conditions: none

Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides.

Step identifiers and dependencies

Rule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S002

S004 · implies

Before

source roles = [0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], Yes]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S003

S005 · implies

Before

source roles = [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [1, 2, 3, 4, 0, 9, 5, 6, 7, 8], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S004

S006 · implies

Before

source roles = [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [2, 3, 4, 0, 1, 8, 9, 5, 6, 7], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S005

S007 · implies

Before

source roles = [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [3, 4, 0, 1, 2, 7, 8, 9, 5, 6], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S006

S008 · implies

Before

source roles = [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [4, 0, 1, 2, 3, 6, 7, 8, 9, 5], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S007

S009 · implies

Before

source roles = [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [Yes, No, No, No, No, Yes, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S008

S010 · implies

Before

source roles = [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [6, 7, 8, 9, 5, 4, 0, 1, 2, 3], [Yes, No, No, No, No, No, Yes, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S009

S011 · implies

Before

source roles = [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [7, 8, 9, 5, 6, 3, 4, 0, 1, 2], [Yes, No, No, No, No, No, No, Yes, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S010

S012 · implies

Before

source roles = [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [8, 9, 5, 6, 7, 2, 3, 4, 0, 1], [Yes, No, No, No, No, No, No, No, Yes, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S011

S013 · implies

Before

source roles = [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

After

derived = [[9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [9, 5, 6, 7, 8, 1, 2, 3, 4, 0], [Yes, No, No, No, No, No, No, No, No, Yes], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S012

S014 · implies

Before

source roles = [Center ACTUAL Z, [0], 0]

After

derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes, [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes]], Yes]

Conditions: none

Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name.

Step identifiers and dependencies

Rule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S013

S015 · implies

Before

source roles = [[0], [[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No], [Yes, No, No, No, No, Yes, No, No, No, No], [Yes, No, No, No, No, No, Yes, No, No, No], [Yes, No, No, No, No, No, No, Yes, No, No], [Yes, No, No, No, No, No, No, No, Yes, No], [Yes, No, No, No, No, No, No, No, No, Yes]]]

After

derived = [Yes, Yes, central z: g*z*g^-1=z for EVERY g]

Conditions: none

Identity commutes with every g;products and inverses of all-g commuting elements do too.Right multiplying gz=zg gives gzg^-1=z,so central elements are fixed pointwise and the center is normal.No characteristic or class-equation claim.

Step identifiers and dependencies

Rule: Center_GENERAL_closure_and_pointwise_fixed_conjugation · Uses: G001, S014

Reverse checks

  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([0], Yes, Yes, Yes, Yes, center_is_a_subgroup_by_complete_identity_product_inverse_audit)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 74197546c6f40af5fa533692c0064fa9ef7cc0baf5496747ab6a3c47f795ef8a
  • candidate record SHA-256: feb16a09348cf5412bddc3b596adfb12120c008c39fe55766f2f1a71285920de
  • semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82
  • question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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