# Question 24544: Left and Right Cosets Must Agree for Every g [Return to catalogue page 123](https://ocme.evemisslab.com/en/questions/page/123/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations/#q-20261015-advance-20261113-r01-finite-normal-subgroup-0069) 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. R0048-24544 · q-20261015-advance-20261113-r01-finite-normal-subgroup-0069 ## Original question and givens Use complete finite Cayley table=[[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]], identity=3 and candidate subset H=[3]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subgroup, coset_rows, all_left_right_equal, certificate). - G001 — Use complete finite Cayley table=[[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]], identity=3 and candidate subset H=[3]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subgroup, coset_rows, all_left_right_equal, 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-normal-subgroup](https://ocme.evemisslab.com/en/mko/mko-finite-normal-subgroup/) ## Step-by-step derivation S001 · impliesBefore source roles = [normal-left-right-coset-audit] After derived = [[subgroup, coset_rows, all_left_right_equal, certificate]] Conditions: none List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [Normality ORIGINAL group, 32, [[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]], 3] 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 Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables. Step identifiers and dependenciesRule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001 S003 · impliesBefore source roles = [Normality ORIGINAL group, 10, FOR EVERY x,y,z in0..n-1] 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, Yes] Conditions: none Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups. Step identifiers and dependenciesRule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002 S004 · impliesBefore source roles = [Normality ORIGINAL group, 3] After derived = [[6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [[0, 6, 3, 3], [1, 5, 3, 3], [2, 4, 3, 3], [3, 3, 3, 3], [4, 2, 3, 3], [5, 1, 3, 3], [6, 0, 3, 3], [7, 9, 3, 3], [8, 8, 3, 3], [9, 7, 3, 3]], Yes] Conditions: none Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises. Step identifiers and dependenciesRule: Group_ALL_two_sided_inverses · Uses: G001, S003 S005 · impliesBefore source roles = [Normality ORIGINAL H, [3], 3] After derived = [Yes, [[3, 3, 3, Yes]], Yes, [[3, 3, Yes]], Yes, Yes] Conditions: none H is nonempty and unique.Check identity membership,all products and every inverse separately;their conjunction qualifies a subgroup.Size divisibility or finiteness alone is insufficient. Step identifiers and dependenciesRule: Subgroup_identity_ALL_products_ALL_inverses_separate · Uses: G001, S004 S006 · impliesBefore source roles = [Normality ORIGINAL H, 0, [3], 6] After derived = [[[3, 0, 6, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S005 S007 · impliesBefore source roles = [Normality ORIGINAL H, 1, [3], 5] After derived = [[[3, 1, 5, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S006 S008 · impliesBefore source roles = [Normality ORIGINAL H, 2, [3], 4] After derived = [[[3, 2, 4, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S007 S009 · impliesBefore source roles = [Normality ORIGINAL H, 3, [3], 3] After derived = [[[3, 3, 3, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S008 S010 · impliesBefore source roles = [Normality ORIGINAL H, 4, [3], 2] After derived = [[[3, 4, 2, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S009 S011 · impliesBefore source roles = [Normality ORIGINAL H, 5, [3], 1] After derived = [[[3, 5, 1, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S010 S012 · impliesBefore source roles = [Normality ORIGINAL H, 6, [3], 0] After derived = [[[3, 6, 0, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S011 S013 · impliesBefore source roles = [Normality ORIGINAL H, 7, [3], 9] After derived = [[[3, 7, 9, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S012 S014 · impliesBefore source roles = [Normality ORIGINAL H, 8, [3], 8] After derived = [[[3, 8, 8, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S013 S015 · impliesBefore source roles = [Normality ORIGINAL H, 9, [3], 7] After derived = [[[3, 9, 7, 3]], [3], Yes] Conditions: none In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators. Step identifiers and dependenciesRule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S014 S016 · impliesBefore source roles = [0, [3]] After derived = [[0], [0], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S015 S017 · impliesBefore source roles = [1, [3]] After derived = [[1], [1], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S016 S018 · impliesBefore source roles = [2, [3]] After derived = [[2], [2], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S017 S019 · impliesBefore source roles = [3, [3]] After derived = [[3], [3], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S018 S020 · impliesBefore source roles = [4, [3]] After derived = [[4], [4], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S019 S021 · impliesBefore source roles = [5, [3]] After derived = [[5], [5], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S020 S022 · impliesBefore source roles = [6, [3]] After derived = [[6], [6], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S021 S023 · impliesBefore source roles = [7, [3]] After derived = [[7], [7], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S022 S024 · impliesBefore source roles = [8, [3]] After derived = [[8], [8], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S023 S025 · impliesBefore source roles = [9, [3]] After derived = [[9], [9], Yes] Conditions: none These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group. Step identifiers and dependenciesRule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S024 S026 · impliesBefore source roles = [[3], Yes, Yes, Yes] After derived = [gHg^-1=H iff gH=Hg for EVERY g] Conditions: none Right multiplication by g is a bijection,turning gHg^-1=H into gH=Hg;right multiply by g inverse for the converse.No quotient multiplication is claimed here. Step identifiers and dependenciesRule: Normality_GENERAL_conjugates_iff_left_right_cosets · Uses: G001, S025 ## Reverse checks - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 ([3], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes], [7, [7], [7], Yes], [8, [8], [8], Yes], [9, [9], [9], Yes]], Yes, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal) 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: 4967f171513391569e09eb77ed94cfd7117029e4b0d96afc542bb53aa2530f99 - candidate record SHA-256: a2ab7897963c179e5dac3404e6da867ec1aac100542359383d5ee008746e455b - semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0 - question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r01-finite-normal-subgroup-0069.json)