# Question 25309: Quotient Products Must Be Representative Independent [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-0034) 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. R0049-25309 · q-20261015-advance-20261114-r02-finite-quotient-group-0034 ## Original question and givens Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18] 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 (cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, certificate). - G001 — Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18] 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 (cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, 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-quotient-group](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-group/) ## Step-by-step derivation S001 · impliesBefore source roles = [quotient-coset-multiplication-table] After derived = [[cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, certificate]] 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, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], 0] After derived = [20, [[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], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19]], 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 = [20, 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, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 8000, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]] 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, 2, 4, 6, 8, 10, 12, 14, 16, 18]] After derived = [[[0, 0, 0, Yes], [0, 2, 2, Yes], [0, 4, 4, Yes], [0, 6, 6, Yes], [0, 8, 8, Yes], [0, 10, 10, Yes], [0, 12, 12, Yes], [0, 14, 14, Yes], [0, 16, 16, Yes], [0, 18, 18, Yes], [2, 0, 2, Yes], [2, 2, 4, Yes], [2, 4, 6, Yes], [2, 6, 8, Yes], [2, 8, 10, Yes], [2, 10, 12, Yes], [2, 12, 14, Yes], [2, 14, 16, Yes], [2, 16, 18, Yes], [2, 18, 0, Yes], [4, 0, 4, Yes], [4, 2, 6, Yes], [4, 4, 8, Yes], [4, 6, 10, Yes], [4, 8, 12, Yes], [4, 10, 14, Yes], [4, 12, 16, Yes], [4, 14, 18, Yes], [4, 16, 0, Yes], [4, 18, 2, Yes], [6, 0, 6, Yes], [6, 2, 8, Yes], [6, 4, 10, Yes], [6, 6, 12, Yes], [6, 8, 14, Yes], [6, 10, 16, Yes], [6, 12, 18, Yes], [6, 14, 0, Yes], [6, 16, 2, Yes], [6, 18, 4, Yes], [8, 0, 8, Yes], [8, 2, 10, Yes], [8, 4, 12, Yes], [8, 6, 14, Yes], [8, 8, 16, Yes], [8, 10, 18, Yes], [8, 12, 0, Yes], [8, 14, 2, Yes], [8, 16, 4, Yes], [8, 18, 6, Yes], [10, 0, 10, Yes], [10, 2, 12, Yes], [10, 4, 14, Yes], [10, 6, 16, Yes], [10, 8, 18, Yes], [10, 10, 0, Yes], [10, 12, 2, Yes], [10, 14, 4, Yes], [10, 16, 6, Yes], [10, 18, 8, Yes], [12, 0, 12, Yes], [12, 2, 14, Yes], [12, 4, 16, Yes], [12, 6, 18, Yes], [12, 8, 0, Yes], [12, 10, 2, Yes], [12, 12, 4, Yes], [12, 14, 6, Yes], [12, 16, 8, Yes], [12, 18, 10, Yes], [14, 0, 14, Yes], [14, 2, 16, Yes], [14, 4, 18, Yes], [14, 6, 0, Yes], [14, 8, 2, Yes], [14, 10, 4, Yes], [14, 12, 6, Yes], [14, 14, 8, Yes], [14, 16, 10, Yes], [14, 18, 12, Yes], [16, 0, 16, Yes], [16, 2, 18, Yes], [16, 4, 0, Yes], [16, 6, 2, Yes], [16, 8, 4, Yes], [16, 10, 6, Yes], [16, 12, 8, Yes], [16, 14, 10, Yes], [16, 16, 12, Yes], [16, 18, 14, Yes], [18, 0, 18, Yes], [18, 2, 0, Yes], [18, 4, 2, Yes], [18, 6, 4, Yes], [18, 8, 6, Yes], [18, 10, 8, Yes], [18, 12, 10, Yes], [18, 14, 12, Yes], [18, 16, 14, Yes], [18, 18, 16, Yes]], [[0, 0, Yes], [2, 18, Yes], [4, 16, Yes], [6, 14, Yes], [8, 12, Yes], [10, 10, Yes], [12, 8, Yes], [14, 6, Yes], [16, 4, Yes], [18, 2, Yes]], Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]]], Yes] 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, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]] After derived = [[[0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 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, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]] After derived = [[[0, 0, 0, 0], [0, 2, 2, 0], [0, 4, 4, 0], [0, 6, 6, 0], [0, 8, 8, 0], [0, 10, 10, 0], [0, 12, 12, 0], [0, 14, 14, 0], [0, 16, 16, 0], [0, 18, 18, 0], [2, 0, 2, 0], [2, 2, 4, 0], [2, 4, 6, 0], [2, 6, 8, 0], [2, 8, 10, 0], [2, 10, 12, 0], [2, 12, 14, 0], [2, 14, 16, 0], [2, 16, 18, 0], [2, 18, 0, 0], [4, 0, 4, 0], [4, 2, 6, 0], [4, 4, 8, 0], [4, 6, 10, 0], [4, 8, 12, 0], [4, 10, 14, 0], [4, 12, 16, 0], [4, 14, 18, 0], [4, 16, 0, 0], [4, 18, 2, 0], [6, 0, 6, 0], [6, 2, 8, 0], [6, 4, 10, 0], [6, 6, 12, 0], [6, 8, 14, 0], [6, 10, 16, 0], [6, 12, 18, 0], [6, 14, 0, 0], [6, 16, 2, 0], [6, 18, 4, 0], [8, 0, 8, 0], [8, 2, 10, 0], [8, 4, 12, 0], [8, 6, 14, 0], [8, 8, 16, 0], [8, 10, 18, 0], [8, 12, 0, 0], [8, 14, 2, 0], [8, 16, 4, 0], [8, 18, 6, 0], [10, 0, 10, 0], [10, 2, 12, 0], [10, 4, 14, 0], [10, 6, 16, 0], [10, 8, 18, 0], [10, 10, 0, 0], [10, 12, 2, 0], [10, 14, 4, 0], [10, 16, 6, 0], [10, 18, 8, 0], [12, 0, 12, 0], [12, 2, 14, 0], [12, 4, 16, 0], [12, 6, 18, 0], [12, 8, 0, 0], [12, 10, 2, 0], [12, 12, 4, 0], [12, 14, 6, 0], [12, 16, 8, 0], [12, 18, 10, 0], [14, 0, 14, 0], [14, 2, 16, 0], [14, 4, 18, 0], [14, 6, 0, 0], [14, 8, 2, 0], [14, 10, 4, 0], [14, 12, 6, 0], [14, 14, 8, 0], [14, 16, 10, 0], [14, 18, 12, 0], [16, 0, 16, 0], [16, 2, 18, 0], [16, 4, 0, 0], [16, 6, 2, 0], [16, 8, 4, 0], [16, 10, 6, 0], [16, 12, 8, 0], [16, 14, 10, 0], [16, 16, 12, 0], [16, 18, 14, 0], [18, 0, 18, 0], [18, 2, 0, 0], [18, 4, 2, 0], [18, 6, 4, 0], [18, 8, 6, 0], [18, 10, 8, 0], [18, 12, 10, 0], [18, 14, 12, 0], [18, 16, 14, 0], [18, 18, 16, 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, 2, 4, 6, 8, 10, 12, 14, 16, 18], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]] After derived = [[[0, 1, 1, 1], [0, 3, 3, 1], [0, 5, 5, 1], [0, 7, 7, 1], [0, 9, 9, 1], [0, 11, 11, 1], [0, 13, 13, 1], [0, 15, 15, 1], [0, 17, 17, 1], [0, 19, 19, 1], [2, 1, 3, 1], [2, 3, 5, 1], [2, 5, 7, 1], [2, 7, 9, 1], [2, 9, 11, 1], [2, 11, 13, 1], [2, 13, 15, 1], [2, 15, 17, 1], [2, 17, 19, 1], [2, 19, 1, 1], [4, 1, 5, 1], [4, 3, 7, 1], [4, 5, 9, 1], [4, 7, 11, 1], [4, 9, 13, 1], [4, 11, 15, 1], [4, 13, 17, 1], [4, 15, 19, 1], [4, 17, 1, 1], [4, 19, 3, 1], [6, 1, 7, 1], [6, 3, 9, 1], [6, 5, 11, 1], [6, 7, 13, 1], [6, 9, 15, 1], [6, 11, 17, 1], [6, 13, 19, 1], [6, 15, 1, 1], [6, 17, 3, 1], [6, 19, 5, 1], [8, 1, 9, 1], [8, 3, 11, 1], [8, 5, 13, 1], [8, 7, 15, 1], [8, 9, 17, 1], [8, 11, 19, 1], [8, 13, 1, 1], [8, 15, 3, 1], [8, 17, 5, 1], [8, 19, 7, 1], [10, 1, 11, 1], [10, 3, 13, 1], [10, 5, 15, 1], [10, 7, 17, 1], [10, 9, 19, 1], [10, 11, 1, 1], [10, 13, 3, 1], [10, 15, 5, 1], [10, 17, 7, 1], [10, 19, 9, 1], [12, 1, 13, 1], [12, 3, 15, 1], [12, 5, 17, 1], [12, 7, 19, 1], [12, 9, 1, 1], [12, 11, 3, 1], [12, 13, 5, 1], [12, 15, 7, 1], [12, 17, 9, 1], [12, 19, 11, 1], [14, 1, 15, 1], [14, 3, 17, 1], [14, 5, 19, 1], [14, 7, 1, 1], [14, 9, 3, 1], [14, 11, 5, 1], [14, 13, 7, 1], [14, 15, 9, 1], [14, 17, 11, 1], [14, 19, 13, 1], [16, 1, 17, 1], [16, 3, 19, 1], [16, 5, 1, 1], [16, 7, 3, 1], [16, 9, 5, 1], [16, 11, 7, 1], [16, 13, 9, 1], [16, 15, 11, 1], [16, 17, 13, 1], [16, 19, 15, 1], [18, 1, 19, 1], [18, 3, 1, 1], [18, 5, 3, 1], [18, 7, 5, 1], [18, 9, 7, 1], [18, 11, 9, 1], [18, 13, 11, 1], [18, 15, 13, 1], [18, 17, 15, 1], [18, 19, 17, 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, S006 S008 · impliesBefore source roles = [1, 0, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]] After derived = [[[1, 0, 1, 1], [1, 2, 3, 1], [1, 4, 5, 1], [1, 6, 7, 1], [1, 8, 9, 1], [1, 10, 11, 1], [1, 12, 13, 1], [1, 14, 15, 1], [1, 16, 17, 1], [1, 18, 19, 1], [3, 0, 3, 1], [3, 2, 5, 1], [3, 4, 7, 1], [3, 6, 9, 1], [3, 8, 11, 1], [3, 10, 13, 1], [3, 12, 15, 1], [3, 14, 17, 1], [3, 16, 19, 1], [3, 18, 1, 1], [5, 0, 5, 1], [5, 2, 7, 1], [5, 4, 9, 1], [5, 6, 11, 1], [5, 8, 13, 1], [5, 10, 15, 1], [5, 12, 17, 1], [5, 14, 19, 1], [5, 16, 1, 1], [5, 18, 3, 1], [7, 0, 7, 1], [7, 2, 9, 1], [7, 4, 11, 1], [7, 6, 13, 1], [7, 8, 15, 1], [7, 10, 17, 1], [7, 12, 19, 1], [7, 14, 1, 1], [7, 16, 3, 1], [7, 18, 5, 1], [9, 0, 9, 1], [9, 2, 11, 1], [9, 4, 13, 1], [9, 6, 15, 1], [9, 8, 17, 1], [9, 10, 19, 1], [9, 12, 1, 1], [9, 14, 3, 1], [9, 16, 5, 1], [9, 18, 7, 1], [11, 0, 11, 1], [11, 2, 13, 1], [11, 4, 15, 1], [11, 6, 17, 1], [11, 8, 19, 1], [11, 10, 1, 1], [11, 12, 3, 1], [11, 14, 5, 1], [11, 16, 7, 1], [11, 18, 9, 1], [13, 0, 13, 1], [13, 2, 15, 1], [13, 4, 17, 1], [13, 6, 19, 1], [13, 8, 1, 1], [13, 10, 3, 1], [13, 12, 5, 1], [13, 14, 7, 1], [13, 16, 9, 1], [13, 18, 11, 1], [15, 0, 15, 1], [15, 2, 17, 1], [15, 4, 19, 1], [15, 6, 1, 1], [15, 8, 3, 1], [15, 10, 5, 1], [15, 12, 7, 1], [15, 14, 9, 1], [15, 16, 11, 1], [15, 18, 13, 1], [17, 0, 17, 1], [17, 2, 19, 1], [17, 4, 1, 1], [17, 6, 3, 1], [17, 8, 5, 1], [17, 10, 7, 1], [17, 12, 9, 1], [17, 14, 11, 1], [17, 16, 13, 1], [17, 18, 15, 1], [19, 0, 19, 1], [19, 2, 1, 1], [19, 4, 3, 1], [19, 6, 5, 1], [19, 8, 7, 1], [19, 10, 9, 1], [19, 12, 11, 1], [19, 14, 13, 1], [19, 16, 15, 1], [19, 18, 17, 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, S007 S009 · impliesBefore source roles = [1, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]] After derived = [[[1, 1, 2, 0], [1, 3, 4, 0], [1, 5, 6, 0], [1, 7, 8, 0], [1, 9, 10, 0], [1, 11, 12, 0], [1, 13, 14, 0], [1, 15, 16, 0], [1, 17, 18, 0], [1, 19, 0, 0], [3, 1, 4, 0], [3, 3, 6, 0], [3, 5, 8, 0], [3, 7, 10, 0], [3, 9, 12, 0], [3, 11, 14, 0], [3, 13, 16, 0], [3, 15, 18, 0], [3, 17, 0, 0], [3, 19, 2, 0], [5, 1, 6, 0], [5, 3, 8, 0], [5, 5, 10, 0], [5, 7, 12, 0], [5, 9, 14, 0], [5, 11, 16, 0], [5, 13, 18, 0], [5, 15, 0, 0], [5, 17, 2, 0], [5, 19, 4, 0], [7, 1, 8, 0], [7, 3, 10, 0], [7, 5, 12, 0], [7, 7, 14, 0], [7, 9, 16, 0], [7, 11, 18, 0], [7, 13, 0, 0], [7, 15, 2, 0], [7, 17, 4, 0], [7, 19, 6, 0], [9, 1, 10, 0], [9, 3, 12, 0], [9, 5, 14, 0], [9, 7, 16, 0], [9, 9, 18, 0], [9, 11, 0, 0], [9, 13, 2, 0], [9, 15, 4, 0], [9, 17, 6, 0], [9, 19, 8, 0], [11, 1, 12, 0], [11, 3, 14, 0], [11, 5, 16, 0], [11, 7, 18, 0], [11, 9, 0, 0], [11, 11, 2, 0], [11, 13, 4, 0], [11, 15, 6, 0], [11, 17, 8, 0], [11, 19, 10, 0], [13, 1, 14, 0], [13, 3, 16, 0], [13, 5, 18, 0], [13, 7, 0, 0], [13, 9, 2, 0], [13, 11, 4, 0], [13, 13, 6, 0], [13, 15, 8, 0], [13, 17, 10, 0], [13, 19, 12, 0], [15, 1, 16, 0], [15, 3, 18, 0], [15, 5, 0, 0], [15, 7, 2, 0], [15, 9, 4, 0], [15, 11, 6, 0], [15, 13, 8, 0], [15, 15, 10, 0], [15, 17, 12, 0], [15, 19, 14, 0], [17, 1, 18, 0], [17, 3, 0, 0], [17, 5, 2, 0], [17, 7, 4, 0], [17, 9, 6, 0], [17, 11, 8, 0], [17, 13, 10, 0], [17, 15, 12, 0], [17, 17, 14, 0], [17, 19, 16, 0], [19, 1, 0, 0], [19, 3, 2, 0], [19, 5, 4, 0], [19, 7, 6, 0], [19, 9, 8, 0], [19, 11, 10, 0], [19, 13, 12, 0], [19, 15, 14, 0], [19, 17, 16, 0], [19, 19, 18, 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, S008 S010 · impliesBefore source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]]] After derived = [[[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], Yes, No] Conditions: none Replacing a by ah and b by bk yields(ah)(bk)=ab(b^-1hb)k in abH,using normality for representative independence.Associativity descends from G,eH is identity and a^-1H inverse.The fixed No withholds automatic commutativity;it does not call a particular cyclic quotient noncommutative or invoke first isomorphism. Step identifiers and dependenciesRule: Quotient_GENERAL_normality_representative_replacement_and_group_laws · Uses: G001, S009 ## Reverse checks - 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, 2, 4, 6, 8, 10, 12, 14, 16, 18], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]], [[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], Yes, No, coset_product_preserves_identity_and_inverse_but_need_not_be_commutative) 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: 7820ae622d33464761721e385f1d06654c05012230a698bea96aabaf98499554 - candidate record SHA-256: bf1734166864d9f28d868cc5324900b40e169a0320f6abf26ba520de395dafbc - 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-0034.json)