# Question 25329: Explicit Representative Dependence Blocks This Quotient Multiplication [Return to catalogue page 127](https://ocme.evemisslab.com/en/questions/page/127/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations/#q-20261015-advance-20261114-r02-finite-quotient-group-0079) 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. R0049-25329 · q-20261015-advance-20261114-r02-finite-quotient-group-0079 ## Original question and givens Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 6] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary). - G001 — Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 6] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary). ## 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-quotient-group](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-group/) ## Step-by-step derivation S001 · impliesBefore source roles = [quotient-nonnormal-boundary] After derived = [[subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary]] Conditions: none List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0] After derived = [12, [[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], [10, 10, 10, 10], [11, 11, 11, 11]], Yes] Conditions: none Validate the complete closed Cayley table and actual identity label,without assuming0 or the old32-element cap. Step identifiers and dependenciesRule: Quotient_complete_group_table_actual_identity · Uses: G001, S001 S003 · impliesBefore source roles = [12, 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, 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]], 1728, [0, 5, 4, 3, 2, 1, 6, 7, 8, 9, 10, 11]] Conditions: none Each pair-matrix entry checks associativity for every z,covering n^3 obligations;each inverse returns the same identity from both sides.This is exhaustive,not sampling. Step identifiers and dependenciesRule: Quotient_ALL_group_associativity_triples_and_two_sided_inverses · Uses: G001, S002 S004 · impliesBefore source roles = [0, [0, 6]] After derived = [[[0, 0, 0, Yes], [0, 6, 6, Yes], [6, 0, 6, Yes], [6, 6, 0, Yes]], [[0, 0, Yes], [6, 6, Yes]], Yes, [[0, [0, 6]], [1, [0, 8]], [2, [0, 10]], [3, [0, 6]], [4, [0, 8]], [5, [0, 10]], [6, [0, 6]], [7, [0, 8]], [8, [0, 10]], [9, [0, 6]], [10, [0, 8]], [11, [0, 10]]], No] Conditions: none First qualify identity,all products and inverse closure as a genuine subgroup,then test conjugation for every g.Positive branches require normality and the boundary requires actual nonnormality,not failed subgroup qualification. Step identifiers and dependenciesRule: Quotient_genuine_subgroup_before_actual_normality · Uses: G001, S003 S005 · impliesBefore source roles = [[0, 6], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]] After derived = [[[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], [0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5], Yes] Conditions: none Every g belongs to gH.If aH meets bH,ah1=bh2 implies b^-1a in H and the complete cosets coincide.Enumerate every g,deduplicate and retain source order.Quotient elements are coset SETS,not least representatives. Step identifiers and dependenciesRule: Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition · Uses: G001, S004 S006 · impliesBefore source roles = [0, 0, [0, 6], [0, 6]] After derived = [[[0, 0, 0, 0], [0, 6, 6, 0], [6, 0, 6, 0], [6, 6, 0, 0]], [0], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S005 S007 · impliesBefore source roles = [0, 1, [0, 6], [1, 7]] After derived = [[[0, 1, 1, 1], [0, 7, 7, 1], [6, 1, 11, 5], [6, 7, 5, 5]], [1, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S006 S008 · impliesBefore source roles = [0, 2, [0, 6], [2, 8]] After derived = [[[0, 2, 2, 2], [0, 8, 8, 2], [6, 2, 10, 4], [6, 8, 4, 4]], [2, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S007 S009 · impliesBefore source roles = [0, 3, [0, 6], [3, 9]] After derived = [[[0, 3, 3, 3], [0, 9, 9, 3], [6, 3, 9, 3], [6, 9, 3, 3]], [3], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S008 S010 · impliesBefore source roles = [0, 4, [0, 6], [4, 10]] After derived = [[[0, 4, 4, 4], [0, 10, 10, 4], [6, 4, 8, 2], [6, 10, 2, 2]], [2, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S009 S011 · impliesBefore source roles = [0, 5, [0, 6], [5, 11]] After derived = [[[0, 5, 5, 5], [0, 11, 11, 5], [6, 5, 7, 1], [6, 11, 1, 1]], [1, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S010 S012 · impliesBefore source roles = [1, 0, [1, 7], [0, 6]] After derived = [[[1, 0, 1, 1], [1, 6, 7, 1], [7, 0, 7, 1], [7, 6, 1, 1]], [1], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S011 S013 · impliesBefore source roles = [1, 1, [1, 7], [1, 7]] After derived = [[[1, 1, 2, 2], [1, 7, 8, 2], [7, 1, 6, 0], [7, 7, 0, 0]], [0, 2], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S012 S014 · impliesBefore source roles = [1, 2, [1, 7], [2, 8]] After derived = [[[1, 2, 3, 3], [1, 8, 9, 3], [7, 2, 11, 5], [7, 8, 5, 5]], [3, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S013 S015 · impliesBefore source roles = [1, 3, [1, 7], [3, 9]] After derived = [[[1, 3, 4, 4], [1, 9, 10, 4], [7, 3, 10, 4], [7, 9, 4, 4]], [4], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S014 S016 · impliesBefore source roles = [1, 4, [1, 7], [4, 10]] After derived = [[[1, 4, 5, 5], [1, 10, 11, 5], [7, 4, 9, 3], [7, 10, 3, 3]], [3, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S015 S017 · impliesBefore source roles = [1, 5, [1, 7], [5, 11]] After derived = [[[1, 5, 0, 0], [1, 11, 6, 0], [7, 5, 8, 2], [7, 11, 2, 2]], [0, 2], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S016 S018 · impliesBefore source roles = [2, 0, [2, 8], [0, 6]] After derived = [[[2, 0, 2, 2], [2, 6, 8, 2], [8, 0, 8, 2], [8, 6, 2, 2]], [2], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S017 S019 · impliesBefore source roles = [2, 1, [2, 8], [1, 7]] After derived = [[[2, 1, 3, 3], [2, 7, 9, 3], [8, 1, 7, 1], [8, 7, 1, 1]], [1, 3], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S018 S020 · impliesBefore source roles = [2, 2, [2, 8], [2, 8]] After derived = [[[2, 2, 4, 4], [2, 8, 10, 4], [8, 2, 6, 0], [8, 8, 0, 0]], [0, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S019 S021 · impliesBefore source roles = [2, 3, [2, 8], [3, 9]] After derived = [[[2, 3, 5, 5], [2, 9, 11, 5], [8, 3, 11, 5], [8, 9, 5, 5]], [5], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S020 S022 · impliesBefore source roles = [2, 4, [2, 8], [4, 10]] After derived = [[[2, 4, 0, 0], [2, 10, 6, 0], [8, 4, 10, 4], [8, 10, 4, 4]], [0, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S021 S023 · impliesBefore source roles = [2, 5, [2, 8], [5, 11]] After derived = [[[2, 5, 1, 1], [2, 11, 7, 1], [8, 5, 9, 3], [8, 11, 3, 3]], [1, 3], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S022 S024 · impliesBefore source roles = [3, 0, [3, 9], [0, 6]] After derived = [[[3, 0, 3, 3], [3, 6, 9, 3], [9, 0, 9, 3], [9, 6, 3, 3]], [3], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S023 S025 · impliesBefore source roles = [3, 1, [3, 9], [1, 7]] After derived = [[[3, 1, 4, 4], [3, 7, 10, 4], [9, 1, 8, 2], [9, 7, 2, 2]], [2, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S024 S026 · impliesBefore source roles = [3, 2, [3, 9], [2, 8]] After derived = [[[3, 2, 5, 5], [3, 8, 11, 5], [9, 2, 7, 1], [9, 8, 1, 1]], [1, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S025 S027 · impliesBefore source roles = [3, 3, [3, 9], [3, 9]] After derived = [[[3, 3, 0, 0], [3, 9, 6, 0], [9, 3, 6, 0], [9, 9, 0, 0]], [0], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S026 S028 · impliesBefore source roles = [3, 4, [3, 9], [4, 10]] After derived = [[[3, 4, 1, 1], [3, 10, 7, 1], [9, 4, 11, 5], [9, 10, 5, 5]], [1, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S027 S029 · impliesBefore source roles = [3, 5, [3, 9], [5, 11]] After derived = [[[3, 5, 2, 2], [3, 11, 8, 2], [9, 5, 10, 4], [9, 11, 4, 4]], [2, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S028 S030 · impliesBefore source roles = [4, 0, [4, 10], [0, 6]] After derived = [[[4, 0, 4, 4], [4, 6, 10, 4], [10, 0, 10, 4], [10, 6, 4, 4]], [4], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S029 S031 · impliesBefore source roles = [4, 1, [4, 10], [1, 7]] After derived = [[[4, 1, 5, 5], [4, 7, 11, 5], [10, 1, 9, 3], [10, 7, 3, 3]], [3, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S030 S032 · impliesBefore source roles = [4, 2, [4, 10], [2, 8]] After derived = [[[4, 2, 0, 0], [4, 8, 6, 0], [10, 2, 8, 2], [10, 8, 2, 2]], [0, 2], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S031 S033 · impliesBefore source roles = [4, 3, [4, 10], [3, 9]] After derived = [[[4, 3, 1, 1], [4, 9, 7, 1], [10, 3, 7, 1], [10, 9, 1, 1]], [1], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S032 S034 · impliesBefore source roles = [4, 4, [4, 10], [4, 10]] After derived = [[[4, 4, 2, 2], [4, 10, 8, 2], [10, 4, 6, 0], [10, 10, 0, 0]], [0, 2], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S033 S035 · impliesBefore source roles = [4, 5, [4, 10], [5, 11]] After derived = [[[4, 5, 3, 3], [4, 11, 9, 3], [10, 5, 11, 5], [10, 11, 5, 5]], [3, 5], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S034 S036 · impliesBefore source roles = [5, 0, [5, 11], [0, 6]] After derived = [[[5, 0, 5, 5], [5, 6, 11, 5], [11, 0, 11, 5], [11, 6, 5, 5]], [5], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S035 S037 · impliesBefore source roles = [5, 1, [5, 11], [1, 7]] After derived = [[[5, 1, 0, 0], [5, 7, 6, 0], [11, 1, 10, 4], [11, 7, 4, 4]], [0, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S036 S038 · impliesBefore source roles = [5, 2, [5, 11], [2, 8]] After derived = [[[5, 2, 1, 1], [5, 8, 7, 1], [11, 2, 9, 3], [11, 8, 3, 3]], [1, 3], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S037 S039 · impliesBefore source roles = [5, 3, [5, 11], [3, 9]] After derived = [[[5, 3, 2, 2], [5, 9, 8, 2], [11, 3, 8, 2], [11, 9, 2, 2]], [2], Yes] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S038 S040 · impliesBefore source roles = [5, 4, [5, 11], [4, 10]] After derived = [[[5, 4, 3, 3], [5, 10, 9, 3], [11, 4, 7, 1], [11, 10, 1, 1]], [1, 3], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S039 S041 · impliesBefore source roles = [5, 5, [5, 11], [5, 11]] After derived = [[[5, 5, 4, 4], [5, 11, 10, 4], [11, 5, 6, 0], [11, 11, 0, 0]], [0, 4], No] Conditions: none For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices. Step identifiers and dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S040 S042 · impliesBefore source roles = [[0, 1, [1, 5], No], [0, 6], [1, 7]] After derived = [[0, 1, 0, 1, 1, 6, 1, 5], No, No] Conditions: none Keep the source first bad ordered pair and left,left2,right,right2 search order.Different representatives of the same two cosets give different product cosets,so multiplication fails to be well-defined.The subgroup and coset partition remain valid;do not invent a quotient table. Step identifiers and dependenciesRule: Quotient_FIRST_bad_pair_exact_source_nested_representative_witness · Uses: G001, S041 ## Reverse checks - 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 ([0, 6], No, [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], [0, 1, 0, 1, 1, 6, 1, 5], No, No, explicit_representative_dependence_blocks_quotient_group_multiplication) 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: c37b5bacb575f1975b1ebcf287b4f3fd7b9b3a005e4242894451a2f8b4327e6e - candidate record SHA-256: 1c7828d8fff5f42e2a25901eb3e4a3f7bc1223b6bc2745ecb2d6b04261b6fa42 - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r02-finite-quotient-group-0079.json)