# Question 25628: The Center Requires Commutation with Every Group Element [Return to catalogue page 129](https://ocme.evemisslab.com/en/questions/page/129/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r02-finite-group-center-foundations/#q-20261015-advance-20261115-r02-finite-group-center-0003) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0050-25628 · q-20261015-advance-20261115-r02-finite-group-center-0003 ## 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 (commutation_rows, center, 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 (commutation_rows, center, 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. - [mko-finite-group-center](https://ocme.evemisslab.com/en/mko/mko-finite-group-center/) ## Step-by-step derivation S001 · impliesBefore source roles = [center-commutation-table] After derived = [[commutation_rows, center, 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore 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 dependenciesRule: Group_complete_table_actual_identity_bound64 · Uses: G001, S001 S003 · impliesBefore 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 dependenciesRule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S002 S004 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S003 S005 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S004 S006 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S005 S007 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S006 S008 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S007 S009 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S008 S010 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S009 S011 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S010 S012 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S011 S013 · impliesBefore 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 dependenciesRule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S012 S014 · impliesBefore 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 dependenciesRule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S013 S015 · impliesBefore 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 dependenciesRule: 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, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, No, No, No, No, No]], [5, [Yes, No, No, No, No, Yes, No, No, No, No]], [6, [Yes, No, No, No, No, No, Yes, No, No, No]], [7, [Yes, No, No, No, No, No, No, Yes, No, No]], [8, [Yes, No, No, No, No, No, No, No, Yes, No]], [9, [Yes, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g) 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: 6128b9b1c92b110ded49ddee151cec65b5fb6570c79d80b6ec58316dce0c6d77 - candidate record SHA-256: 27c3fac66787bca4f17d1414eea51d5b2bc54e9aeac5cf993b9465a0dfa9dc7a - semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82 - question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r02-finite-group-center-0003.json)