# Question 10975: Explicit Subgroup Closure, Identity and Inverse Obligations [Return to catalogue page 55](https://ocme.evemisslab.com/en/questions/page/55/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations/#q-20261013-advance-20261023-r03-subgroup-homomorphism-0050) 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. R0020-10975 · q-20261013-advance-20261023-r03-subgroup-homomorphism-0050 ## Original question and givens Use complete qualified finite group Cayley tables with carrier labels 0 through size−1 and the given row/column order. Every group axiom is a premise already checked on the whole table. Results concern these finite groups; a homomorphism need not be injective or an isomorphism. size=6, table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 0, 5, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 1, 2, 4], [4, 5, 1, 2, 3, 0], [5, 2, 3, 4, 0, 1]], nonempty subset=[0, 1, 3] normalized as a set. Give (size,subset,group identity,product closed?,contains identity?,inverse closed?,all subgroup checks?). - G001 — Use complete qualified finite group Cayley tables with carrier labels 0 through size−1 and the given row/column order. Every group axiom is a premise already checked on the whole table. Results concern these finite groups; a homomorphism need not be injective or an isomorphism. size=6, table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 0, 5, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 1, 2, 4], [4, 5, 1, 2, 3, 0], [5, 2, 3, 4, 0, 1]], nonempty subset=[0, 1, 3] normalized as a set. Give (size,subset,group identity,product closed?,contains identity?,inverse closed?,all subgroup checks?). ## 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-subgroup-homomorphism](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/) ## Step-by-step derivation S001 · impliesBefore group=ambient;carrier Fin(6);complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 0, 5, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 1, 2, 4], [4, 5, 1, 2, 3, 0], [5, 2, 3, 4, 0, 1]] After typed operation closure=Yes;all associativity failures=[] Conditions: none Type the complete operation table and check every associativity triple,not only cells used by a subset or map.These tasks require qualified finite groups. Step identifiers and dependenciesRule: complete_group_table_closure_associativity · Uses: G001 S002 · impliesBefore group=ambient;candidate identity=0;left products row=[0, 1, 2, 3, 4, 5];right products column=[0, 1, 2, 3, 4, 5] After both sides equal carrier labels=Yes Conditions: none Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0. Step identifiers and dependenciesRule: derive_two_sided_group_identity · Uses: G001, S001 S003 · impliesBefore group=ambient;candidate identity=1;left products row=[1, 3, 4, 0, 5, 2];right products column=[1, 3, 4, 0, 5, 2] After both sides equal carrier labels=No Conditions: none Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0. Step identifiers and dependenciesRule: derive_two_sided_group_identity · Uses: G001, S002 S004 · impliesBefore group=ambient;candidate identity=2;left products row=[2, 4, 0, 5, 1, 3];right products column=[2, 4, 0, 5, 1, 3] After both sides equal carrier labels=No Conditions: none Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0. Step identifiers and dependenciesRule: derive_two_sided_group_identity · Uses: G001, S003 S005 · impliesBefore group=ambient;candidate identity=3;left products row=[3, 0, 5, 1, 2, 4];right products column=[3, 0, 5, 1, 2, 4] After both sides equal carrier labels=No Conditions: none Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0. Step identifiers and dependenciesRule: derive_two_sided_group_identity · Uses: G001, S004 S006 · impliesBefore group=ambient;candidate identity=4;left products row=[4, 5, 1, 2, 3, 0];right products column=[4, 5, 1, 2, 3, 0] After both sides equal carrier labels=No Conditions: none Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0. Step identifiers and dependenciesRule: derive_two_sided_group_identity · Uses: G001, S005 S007 · impliesBefore group=ambient;candidate identity=5;left products row=[5, 2, 3, 4, 0, 1];right products column=[5, 2, 3, 4, 0, 1] After both sides equal carrier labels=No Conditions: none Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0. Step identifiers and dependenciesRule: derive_two_sided_group_identity · Uses: G001, S006 S008 · impliesBefore group=ambient;element=0;derived identity=0;two-sided inverse candidates=[0] After inverse=0 Conditions: none Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse. Step identifiers and dependenciesRule: derive_group_inverses_from_actual_identity · Uses: G001, S007 S009 · impliesBefore group=ambient;element=1;derived identity=0;two-sided inverse candidates=[3] After inverse=3 Conditions: none Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse. Step identifiers and dependenciesRule: derive_group_inverses_from_actual_identity · Uses: G001, S008 S010 · impliesBefore group=ambient;element=2;derived identity=0;two-sided inverse candidates=[2] After inverse=2 Conditions: none Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse. Step identifiers and dependenciesRule: derive_group_inverses_from_actual_identity · Uses: G001, S009 S011 · impliesBefore group=ambient;element=3;derived identity=0;two-sided inverse candidates=[1] After inverse=1 Conditions: none Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse. Step identifiers and dependenciesRule: derive_group_inverses_from_actual_identity · Uses: G001, S010 S012 · impliesBefore group=ambient;element=4;derived identity=0;two-sided inverse candidates=[5] After inverse=5 Conditions: none Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse. Step identifiers and dependenciesRule: derive_group_inverses_from_actual_identity · Uses: G001, S011 S013 · impliesBefore group=ambient;element=5;derived identity=0;two-sided inverse candidates=[4] After inverse=4 Conditions: none Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse. Step identifiers and dependenciesRule: derive_group_inverses_from_actual_identity · Uses: G001, S012 S014 · impliesBefore subset=[0, 1, 3];ordered pair=[0, 0];ambient product=0 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S013 S015 · impliesBefore subset=[0, 1, 3];ordered pair=[0, 1];ambient product=1 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S014 S016 · impliesBefore subset=[0, 1, 3];ordered pair=[0, 3];ambient product=3 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S015 S017 · impliesBefore subset=[0, 1, 3];ordered pair=[1, 0];ambient product=1 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S016 S018 · impliesBefore subset=[0, 1, 3];ordered pair=[1, 1];ambient product=3 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S017 S019 · impliesBefore subset=[0, 1, 3];ordered pair=[1, 3];ambient product=0 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S018 S020 · impliesBefore subset=[0, 1, 3];ordered pair=[3, 0];ambient product=3 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S019 S021 · impliesBefore subset=[0, 1, 3];ordered pair=[3, 1];ambient product=0 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S020 S022 · impliesBefore subset=[0, 1, 3];ordered pair=[3, 3];ambient product=1 After product stays in subset=Yes Conditions: none Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required. Step identifiers and dependenciesRule: subset_every_ordered_product_closure · Uses: G001, S021 S023 · impliesBefore element=0;ambient inverse=0;subset=[0, 1, 3] After inverse retained=Yes Conditions: none Check membership of each inverse derived from the full group,not the element itself or a negated label. Step identifiers and dependenciesRule: subgroup_each_actual_inverse_membership · Uses: G001, S022 S024 · impliesBefore element=1;ambient inverse=3;subset=[0, 1, 3] After inverse retained=Yes Conditions: none Check membership of each inverse derived from the full group,not the element itself or a negated label. Step identifiers and dependenciesRule: subgroup_each_actual_inverse_membership · Uses: G001, S023 S025 · impliesBefore element=3;ambient inverse=1;subset=[0, 1, 3] After inverse retained=Yes Conditions: none Check membership of each inverse derived from the full group,not the element itself or a negated label. Step identifiers and dependenciesRule: subgroup_each_actual_inverse_membership · Uses: G001, S024 S026 · impliesBefore closed=Yes;actual identity=0;identity retained=Yes;inverse closed=Yes After all explicit subgroup obligations=Yes Conditions: none Retain the protocol's three explicit closure,identity and inverse judgments,then take their conjunction. Step identifiers and dependenciesRule: subgroup_explicit_joint_audit · 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 (6, [0, 1, 3], 0, Yes, Yes, Yes, Yes) 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: c4a2053aac02887388e706fdbe49752e0f8c012d0effdeaef4acced8c7278836 - candidate record SHA-256: 9872d65f1c848562d94329ff1fd2a3b5c277723751d0c5266844f50cfd8989f8 - semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f - question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r03-subgroup-homomorphism-0050.json)