# Question 10902: Operation Preservation for Every Source Pair [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-0052) 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. R0020-10902 · q-20261013-advance-20261023-r03-subgroup-homomorphism-0052 ## 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. source size=6, source table=[[2, 0, 3, 4, 5, 1], [0, 1, 2, 3, 4, 5], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3], [1, 5, 0, 2, 3, 4]], target size=3, target table=[[2, 0, 1], [0, 1, 2], [1, 2, 0]], total mapping=[0, 1, 2, 1, 0, 2] in source-label order. Give (source size,target size,mapping,every bad [a,b] pair,homomorphism?). A bad pair violates f(a*b)=f(a)*f(b); use nested increasing source-label order. - 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. source size=6, source table=[[2, 0, 3, 4, 5, 1], [0, 1, 2, 3, 4, 5], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3], [1, 5, 0, 2, 3, 4]], target size=3, target table=[[2, 0, 1], [0, 1, 2], [1, 2, 0]], total mapping=[0, 1, 2, 1, 0, 2] in source-label order. Give (source size,target size,mapping,every bad [a,b] pair,homomorphism?). A bad pair violates f(a*b)=f(a)*f(b); use nested increasing source-label order. ## 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=source;carrier Fin(6);complete Cayley table=[[2, 0, 3, 4, 5, 1], [0, 1, 2, 3, 4, 5], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3], [1, 5, 0, 2, 3, 4]] 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=source;candidate identity=0;left products row=[2, 0, 3, 4, 5, 1];right products column=[2, 0, 3, 4, 5, 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, S001 S003 · impliesBefore group=source;candidate identity=1;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, S002 S004 · impliesBefore group=source;candidate identity=2;left products row=[3, 2, 4, 5, 1, 0];right products column=[3, 2, 4, 5, 1, 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, S003 S005 · impliesBefore group=source;candidate identity=3;left products row=[4, 3, 5, 1, 0, 2];right products column=[4, 3, 5, 1, 0, 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, S004 S006 · impliesBefore group=source;candidate identity=4;left products row=[5, 4, 1, 0, 2, 3];right products column=[5, 4, 1, 0, 2, 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, S005 S007 · impliesBefore group=source;candidate identity=5;left products row=[1, 5, 0, 2, 3, 4];right products column=[1, 5, 0, 2, 3, 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, S006 S008 · impliesBefore group=source;element=0;derived identity=1;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, S007 S009 · impliesBefore group=source;element=1;derived identity=1;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, S008 S010 · impliesBefore group=source;element=2;derived identity=1;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, S009 S011 · impliesBefore group=source;element=3;derived identity=1;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, S010 S012 · impliesBefore group=source;element=4;derived identity=1;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, S011 S013 · impliesBefore group=source;element=5;derived identity=1;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, S012 S014 · impliesBefore group=target;carrier Fin(3);complete Cayley table=[[2, 0, 1], [0, 1, 2], [1, 2, 0]] 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, S013 S015 · impliesBefore group=target;candidate identity=0;left products row=[2, 0, 1];right products column=[2, 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, S014 S016 · impliesBefore group=target;candidate identity=1;left products row=[0, 1, 2];right products column=[0, 1, 2] 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, S015 S017 · impliesBefore group=target;candidate identity=2;left products row=[1, 2, 0];right products column=[1, 2, 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, S016 S018 · impliesBefore group=target;element=0;derived identity=1;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, S017 S019 · impliesBefore group=target;element=1;derived identity=1;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, S018 S020 · impliesBefore group=target;element=2;derived identity=1;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, S019 S021 · impliesBefore source size=6;target size=3;whole mapping=[0, 1, 2, 1, 0, 2] After all original mapping cells typed before product preservation Conditions: none Type the entire total mapping in original source-index order before checking all products.Do not fill missing values from partial correct data. Step identifiers and dependenciesRule: whole_group_mapping_type_guard · Uses: G001, S020 S022 · impliesBefore source pair=[0, 0];source product=2;mapped operands=[0, 0] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S021 S023 · impliesBefore source pair=[0, 1];source product=0;mapped operands=[0, 1] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S022 S024 · impliesBefore source pair=[0, 2];source product=3;mapped operands=[0, 2] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S023 S025 · impliesBefore source pair=[0, 3];source product=4;mapped operands=[0, 1] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S024 S026 · impliesBefore source pair=[0, 4];source product=5;mapped operands=[0, 0] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S025 S027 · impliesBefore source pair=[0, 5];source product=1;mapped operands=[0, 2] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S026 S028 · impliesBefore source pair=[1, 0];source product=0;mapped operands=[1, 0] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S027 S029 · impliesBefore source pair=[1, 1];source product=1;mapped operands=[1, 1] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S028 S030 · impliesBefore source pair=[1, 2];source product=2;mapped operands=[1, 2] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S029 S031 · impliesBefore source pair=[1, 3];source product=3;mapped operands=[1, 1] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S030 S032 · impliesBefore source pair=[1, 4];source product=4;mapped operands=[1, 0] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S031 S033 · impliesBefore source pair=[1, 5];source product=5;mapped operands=[1, 2] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S032 S034 · impliesBefore source pair=[2, 0];source product=3;mapped operands=[2, 0] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S033 S035 · impliesBefore source pair=[2, 1];source product=2;mapped operands=[2, 1] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S034 S036 · impliesBefore source pair=[2, 2];source product=4;mapped operands=[2, 2] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S035 S037 · impliesBefore source pair=[2, 3];source product=5;mapped operands=[2, 1] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S036 S038 · impliesBefore source pair=[2, 4];source product=1;mapped operands=[2, 0] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S037 S039 · impliesBefore source pair=[2, 5];source product=0;mapped operands=[2, 2] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S038 S040 · impliesBefore source pair=[3, 0];source product=4;mapped operands=[1, 0] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S039 S041 · impliesBefore source pair=[3, 1];source product=3;mapped operands=[1, 1] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S040 S042 · impliesBefore source pair=[3, 2];source product=5;mapped operands=[1, 2] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S041 S043 · impliesBefore source pair=[3, 3];source product=1;mapped operands=[1, 1] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S042 S044 · impliesBefore source pair=[3, 4];source product=0;mapped operands=[1, 0] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S043 S045 · impliesBefore source pair=[3, 5];source product=2;mapped operands=[1, 2] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S044 S046 · impliesBefore source pair=[4, 0];source product=5;mapped operands=[0, 0] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S045 S047 · impliesBefore source pair=[4, 1];source product=4;mapped operands=[0, 1] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S046 S048 · impliesBefore source pair=[4, 2];source product=1;mapped operands=[0, 2] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S047 S049 · impliesBefore source pair=[4, 3];source product=0;mapped operands=[0, 1] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S048 S050 · impliesBefore source pair=[4, 4];source product=2;mapped operands=[0, 0] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S049 S051 · impliesBefore source pair=[4, 5];source product=3;mapped operands=[0, 2] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S050 S052 · impliesBefore source pair=[5, 0];source product=1;mapped operands=[2, 0] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S051 S053 · impliesBefore source pair=[5, 1];source product=5;mapped operands=[2, 1] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S052 S054 · impliesBefore source pair=[5, 2];source product=0;mapped operands=[2, 2] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S053 S055 · impliesBefore source pair=[5, 3];source product=2;mapped operands=[2, 1] After f(source product)=2;target product=2;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S054 S056 · impliesBefore source pair=[5, 4];source product=3;mapped operands=[2, 0] After f(source product)=1;target product=1;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S055 S057 · impliesBefore source pair=[5, 5];source product=4;mapped operands=[2, 2] After f(source product)=0;target product=0;equal=Yes Conditions: none The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism. Step identifiers and dependenciesRule: homomorphism_every_source_product_pair · Uses: G001, S056 S058 · impliesBefore all bad source ordered pairs=[] After homomorphism=Yes Conditions: none Preservation for all ordered source pairs is required.One failure supplies a counterexample;one success does not establish the universal condition. Step identifiers and dependenciesRule: whole_homomorphism_audit · Uses: G001, S057 ## Reverse checks - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, 3, [0, 1, 2, 1, 0, 2], [], 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: 501f57137e30713a66422c9c725f7490834e3a3203d8f3df48fef4aa2fdae106 - candidate record SHA-256: 9d41f958b507f5d8e181031a60898987a3ef64df6d2a2925c8d9bbc25f9da185 - 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-0052.json)