# Question 25737: First Isomorphism Reaches the Image, Not Automatically the Whole Codomain [Return to catalogue page 129](https://ocme.evemisslab.com/en/questions/page/129/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/#q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0087) 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. R0050-25737 · q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0087 ## Original question and givens Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary). - G001 — Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, 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-first-isomorphism-theorem-finite-groups](https://ocme.evemisslab.com/en/mko/mko-first-isomorphism-theorem-finite-groups/) ## Step-by-step derivation S001 · impliesBefore source roles = [iso1-image-codomain-boundary] After derived = [[image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary]] Conditions: none List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [Iso1 ORIGINAL SOURCE, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], 0] After derived = [13, [[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]], Yes] Conditions: none Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0. Step identifiers and dependenciesRule: Group_complete_table_actual_identity_bound64 · Uses: G001, S001 S003 · impliesBefore source roles = [Iso1 ORIGINAL SOURCE, 13, 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]], 2197, [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]] Conditions: none Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides. Step identifiers and dependenciesRule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S002 S004 · impliesBefore source roles = [Iso1 ORIGINAL TARGET, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], 0] After derived = [26, [[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], [20, 20, 20, 20], [21, 21, 21, 21], [22, 22, 22, 22], [23, 23, 23, 23], [24, 24, 24, 24], [25, 25, 25, 25]], Yes] Conditions: none Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0. Step identifiers and dependenciesRule: Group_complete_table_actual_identity_bound64 · Uses: G001, S003 S005 · impliesBefore source roles = [Iso1 ORIGINAL TARGET, 26, 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, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 17576, [0, 25, 24, 23, 22, 21, 20, 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 every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides. Step identifiers and dependenciesRule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S004 S006 · impliesBefore source roles = [Iso1 ACTUAL kernel, [0], 0] After derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes, [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes]], Yes] Conditions: none Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name. Step identifiers and dependenciesRule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S005 S007 · impliesBefore source roles = [Iso1 ACTUAL image, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 0] After derived = [Yes, [[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], [0, 20, 20, Yes], [0, 22, 22, Yes], [0, 24, 24, 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, 20, Yes], [2, 20, 22, Yes], [2, 22, 24, Yes], [2, 24, 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, 20, Yes], [4, 18, 22, Yes], [4, 20, 24, Yes], [4, 22, 0, Yes], [4, 24, 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, 20, Yes], [6, 16, 22, Yes], [6, 18, 24, Yes], [6, 20, 0, Yes], [6, 22, 2, Yes], [6, 24, 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, 20, Yes], [8, 14, 22, Yes], [8, 16, 24, Yes], [8, 18, 0, Yes], [8, 20, 2, Yes], [8, 22, 4, Yes], [8, 24, 6, Yes], [10, 0, 10, Yes], [10, 2, 12, Yes], [10, 4, 14, Yes], [10, 6, 16, Yes], [10, 8, 18, Yes], [10, 10, 20, Yes], [10, 12, 22, Yes], [10, 14, 24, Yes], [10, 16, 0, Yes], [10, 18, 2, Yes], [10, 20, 4, Yes], [10, 22, 6, Yes], [10, 24, 8, Yes], [12, 0, 12, Yes], [12, 2, 14, Yes], [12, 4, 16, Yes], [12, 6, 18, Yes], [12, 8, 20, Yes], [12, 10, 22, Yes], [12, 12, 24, Yes], [12, 14, 0, Yes], [12, 16, 2, Yes], [12, 18, 4, Yes], [12, 20, 6, Yes], [12, 22, 8, Yes], [12, 24, 10, Yes], [14, 0, 14, Yes], [14, 2, 16, Yes], [14, 4, 18, Yes], [14, 6, 20, Yes], [14, 8, 22, Yes], [14, 10, 24, Yes], [14, 12, 0, Yes], [14, 14, 2, Yes], [14, 16, 4, Yes], [14, 18, 6, Yes], [14, 20, 8, Yes], [14, 22, 10, Yes], [14, 24, 12, Yes], [16, 0, 16, Yes], [16, 2, 18, Yes], [16, 4, 20, Yes], [16, 6, 22, Yes], [16, 8, 24, Yes], [16, 10, 0, Yes], [16, 12, 2, Yes], [16, 14, 4, Yes], [16, 16, 6, Yes], [16, 18, 8, Yes], [16, 20, 10, Yes], [16, 22, 12, Yes], [16, 24, 14, Yes], [18, 0, 18, Yes], [18, 2, 20, Yes], [18, 4, 22, Yes], [18, 6, 24, Yes], [18, 8, 0, Yes], [18, 10, 2, Yes], [18, 12, 4, Yes], [18, 14, 6, Yes], [18, 16, 8, Yes], [18, 18, 10, Yes], [18, 20, 12, Yes], [18, 22, 14, Yes], [18, 24, 16, Yes], [20, 0, 20, Yes], [20, 2, 22, Yes], [20, 4, 24, Yes], [20, 6, 0, Yes], [20, 8, 2, Yes], [20, 10, 4, Yes], [20, 12, 6, Yes], [20, 14, 8, Yes], [20, 16, 10, Yes], [20, 18, 12, Yes], [20, 20, 14, Yes], [20, 22, 16, Yes], [20, 24, 18, Yes], [22, 0, 22, Yes], [22, 2, 24, Yes], [22, 4, 0, Yes], [22, 6, 2, Yes], [22, 8, 4, Yes], [22, 10, 6, Yes], [22, 12, 8, Yes], [22, 14, 10, Yes], [22, 16, 12, Yes], [22, 18, 14, Yes], [22, 20, 16, Yes], [22, 22, 18, Yes], [22, 24, 20, Yes], [24, 0, 24, Yes], [24, 2, 0, Yes], [24, 4, 2, Yes], [24, 6, 4, Yes], [24, 8, 6, Yes], [24, 10, 8, Yes], [24, 12, 10, Yes], [24, 14, 12, Yes], [24, 16, 14, Yes], [24, 18, 16, Yes], [24, 20, 18, Yes], [24, 22, 20, Yes], [24, 24, 22, Yes]], Yes, [[0, 0, Yes], [2, 24, Yes], [4, 22, Yes], [6, 20, Yes], [8, 18, Yes], [10, 16, Yes], [12, 14, Yes], [14, 12, Yes], [16, 10, Yes], [18, 8, Yes], [20, 6, Yes], [22, 4, Yes], [24, 2, Yes]], Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes]], Yes] Conditions: none Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name. Step identifiers and dependenciesRule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S006 S008 · impliesBefore source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 0, 0] After derived = [[[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 6, 6, Yes], [0, 4, 8, 8, Yes], [0, 5, 10, 10, Yes], [0, 6, 12, 12, Yes], [0, 7, 14, 14, Yes], [0, 8, 16, 16, Yes], [0, 9, 18, 18, Yes], [0, 10, 20, 20, Yes], [0, 11, 22, 22, Yes], [0, 12, 24, 24, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 6, 6, Yes], [1, 3, 8, 8, Yes], [1, 4, 10, 10, Yes], [1, 5, 12, 12, Yes], [1, 6, 14, 14, Yes], [1, 7, 16, 16, Yes], [1, 8, 18, 18, Yes], [1, 9, 20, 20, Yes], [1, 10, 22, 22, Yes], [1, 11, 24, 24, Yes], [1, 12, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 6, 6, Yes], [2, 2, 8, 8, Yes], [2, 3, 10, 10, Yes], [2, 4, 12, 12, Yes], [2, 5, 14, 14, Yes], [2, 6, 16, 16, Yes], [2, 7, 18, 18, Yes], [2, 8, 20, 20, Yes], [2, 9, 22, 22, Yes], [2, 10, 24, 24, Yes], [2, 11, 0, 0, Yes], [2, 12, 2, 2, Yes], [3, 0, 6, 6, Yes], [3, 1, 8, 8, Yes], [3, 2, 10, 10, Yes], [3, 3, 12, 12, Yes], [3, 4, 14, 14, Yes], [3, 5, 16, 16, Yes], [3, 6, 18, 18, Yes], [3, 7, 20, 20, Yes], [3, 8, 22, 22, Yes], [3, 9, 24, 24, Yes], [3, 10, 0, 0, Yes], [3, 11, 2, 2, Yes], [3, 12, 4, 4, Yes], [4, 0, 8, 8, Yes], [4, 1, 10, 10, Yes], [4, 2, 12, 12, Yes], [4, 3, 14, 14, Yes], [4, 4, 16, 16, Yes], [4, 5, 18, 18, Yes], [4, 6, 20, 20, Yes], [4, 7, 22, 22, Yes], [4, 8, 24, 24, Yes], [4, 9, 0, 0, Yes], [4, 10, 2, 2, Yes], [4, 11, 4, 4, Yes], [4, 12, 6, 6, Yes], [5, 0, 10, 10, Yes], [5, 1, 12, 12, Yes], [5, 2, 14, 14, Yes], [5, 3, 16, 16, Yes], [5, 4, 18, 18, Yes], [5, 5, 20, 20, Yes], [5, 6, 22, 22, Yes], [5, 7, 24, 24, Yes], [5, 8, 0, 0, Yes], [5, 9, 2, 2, Yes], [5, 10, 4, 4, Yes], [5, 11, 6, 6, Yes], [5, 12, 8, 8, Yes], [6, 0, 12, 12, Yes], [6, 1, 14, 14, Yes], [6, 2, 16, 16, Yes], [6, 3, 18, 18, Yes], [6, 4, 20, 20, Yes], [6, 5, 22, 22, Yes], [6, 6, 24, 24, Yes], [6, 7, 0, 0, Yes], [6, 8, 2, 2, Yes], [6, 9, 4, 4, Yes], [6, 10, 6, 6, Yes], [6, 11, 8, 8, Yes], [6, 12, 10, 10, Yes], [7, 0, 14, 14, Yes], [7, 1, 16, 16, Yes], [7, 2, 18, 18, Yes], [7, 3, 20, 20, Yes], [7, 4, 22, 22, Yes], [7, 5, 24, 24, Yes], [7, 6, 0, 0, Yes], [7, 7, 2, 2, Yes], [7, 8, 4, 4, Yes], [7, 9, 6, 6, Yes], [7, 10, 8, 8, Yes], [7, 11, 10, 10, Yes], [7, 12, 12, 12, Yes], [8, 0, 16, 16, Yes], [8, 1, 18, 18, Yes], [8, 2, 20, 20, Yes], [8, 3, 22, 22, Yes], [8, 4, 24, 24, Yes], [8, 5, 0, 0, Yes], [8, 6, 2, 2, Yes], [8, 7, 4, 4, Yes], [8, 8, 6, 6, Yes], [8, 9, 8, 8, Yes], [8, 10, 10, 10, Yes], [8, 11, 12, 12, Yes], [8, 12, 14, 14, Yes], [9, 0, 18, 18, Yes], [9, 1, 20, 20, Yes], [9, 2, 22, 22, Yes], [9, 3, 24, 24, Yes], [9, 4, 0, 0, Yes], [9, 5, 2, 2, Yes], [9, 6, 4, 4, Yes], [9, 7, 6, 6, Yes], [9, 8, 8, 8, Yes], [9, 9, 10, 10, Yes], [9, 10, 12, 12, Yes], [9, 11, 14, 14, Yes], [9, 12, 16, 16, Yes], [10, 0, 20, 20, Yes], [10, 1, 22, 22, Yes], [10, 2, 24, 24, Yes], [10, 3, 0, 0, Yes], [10, 4, 2, 2, Yes], [10, 5, 4, 4, Yes], [10, 6, 6, 6, Yes], [10, 7, 8, 8, Yes], [10, 8, 10, 10, Yes], [10, 9, 12, 12, Yes], [10, 10, 14, 14, Yes], [10, 11, 16, 16, Yes], [10, 12, 18, 18, Yes], [11, 0, 22, 22, Yes], [11, 1, 24, 24, Yes], [11, 2, 0, 0, Yes], [11, 3, 2, 2, Yes], [11, 4, 4, 4, Yes], [11, 5, 6, 6, Yes], [11, 6, 8, 8, Yes], [11, 7, 10, 10, Yes], [11, 8, 12, 12, Yes], [11, 9, 14, 14, Yes], [11, 10, 16, 16, Yes], [11, 11, 18, 18, Yes], [11, 12, 20, 20, Yes], [12, 0, 24, 24, Yes], [12, 1, 0, 0, Yes], [12, 2, 2, 2, Yes], [12, 3, 4, 4, Yes], [12, 4, 6, 6, Yes], [12, 5, 8, 8, Yes], [12, 6, 10, 10, Yes], [12, 7, 12, 12, Yes], [12, 8, 14, 14, Yes], [12, 9, 16, 16, Yes], [12, 10, 18, 18, Yes], [12, 11, 20, 20, Yes], [12, 12, 22, 22, Yes]], Yes, null, [0], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes, Yes] Conditions: none Audit every homomorphism pair and the full target-identity preimage/image.Qualification allows a bad map and reports actual properties:homFalse can coexist with normal kernel or subgroup image.An empty kernel is not replaced by an identity fallback. Step identifiers and dependenciesRule: Iso1_ALL_product_laws_ACTUAL_kernel_image_properties_before_theorem · Uses: G001, S007 S009 · impliesBefore source roles = [[0], Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]] After derived = [[[0], [1], [10], [11], [12], [2], [3], [4], [5], [6], [7], [8], [9]], [0, 1, 5, 6, 7, 8, 9, 10, 11, 12, 2, 3, 4], [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 5, 3, 4, 0, 6, 7, 8, 9, 10, 11, 12, 2], [2, 3, 10, 11, 12, 4, 0, 1, 5, 6, 7, 8, 9], [3, 4, 11, 12, 2, 0, 1, 5, 6, 7, 8, 9, 10], [4, 0, 12, 2, 3, 1, 5, 6, 7, 8, 9, 10, 11], [5, 6, 4, 0, 1, 7, 8, 9, 10, 11, 12, 2, 3], [6, 7, 0, 1, 5, 8, 9, 10, 11, 12, 2, 3, 4], [7, 8, 1, 5, 6, 9, 10, 11, 12, 2, 3, 4, 0], [8, 9, 5, 6, 7, 10, 11, 12, 2, 3, 4, 0, 1], [9, 10, 6, 7, 8, 11, 12, 2, 3, 4, 0, 1, 5], [10, 11, 7, 8, 9, 12, 2, 3, 4, 0, 1, 5, 6], [11, 12, 8, 9, 10, 2, 3, 4, 0, 1, 5, 6, 7], [12, 2, 9, 10, 11, 3, 4, 0, 1, 5, 6, 7, 8]], [[0, 0, [0], Yes], [0, 1, [1], Yes], [0, 2, [2], Yes], [0, 3, [3], Yes], [0, 4, [4], Yes], [0, 5, [5], Yes], [0, 6, [6], Yes], [0, 7, [7], Yes], [0, 8, [8], Yes], [0, 9, [9], Yes], [0, 10, [10], Yes], [0, 11, [11], Yes], [0, 12, [12], Yes], [1, 0, [1], Yes], [1, 1, [5], Yes], [1, 2, [3], Yes], [1, 3, [4], Yes], [1, 4, [0], Yes], [1, 5, [6], Yes], [1, 6, [7], Yes], [1, 7, [8], Yes], [1, 8, [9], Yes], [1, 9, [10], Yes], [1, 10, [11], Yes], [1, 11, [12], Yes], [1, 12, [2], Yes], [2, 0, [2], Yes], [2, 1, [3], Yes], [2, 2, [10], Yes], [2, 3, [11], Yes], [2, 4, [12], Yes], [2, 5, [4], Yes], [2, 6, [0], Yes], [2, 7, [1], Yes], [2, 8, [5], Yes], [2, 9, [6], Yes], [2, 10, [7], Yes], [2, 11, [8], Yes], [2, 12, [9], Yes], [3, 0, [3], Yes], [3, 1, [4], Yes], [3, 2, [11], Yes], [3, 3, [12], Yes], [3, 4, [2], Yes], [3, 5, [0], Yes], [3, 6, [1], Yes], [3, 7, [5], Yes], [3, 8, [6], Yes], [3, 9, [7], Yes], [3, 10, [8], Yes], [3, 11, [9], Yes], [3, 12, [10], Yes], [4, 0, [4], Yes], [4, 1, [0], Yes], [4, 2, [12], Yes], [4, 3, [2], Yes], [4, 4, [3], Yes], [4, 5, [1], Yes], [4, 6, [5], Yes], [4, 7, [6], Yes], [4, 8, [7], Yes], [4, 9, [8], Yes], [4, 10, [9], Yes], [4, 11, [10], Yes], [4, 12, [11], Yes], [5, 0, [5], Yes], [5, 1, [6], Yes], [5, 2, [4], Yes], [5, 3, [0], Yes], [5, 4, [1], Yes], [5, 5, [7], Yes], [5, 6, [8], Yes], [5, 7, [9], Yes], [5, 8, [10], Yes], [5, 9, [11], Yes], [5, 10, [12], Yes], [5, 11, [2], Yes], [5, 12, [3], Yes], [6, 0, [6], Yes], [6, 1, [7], Yes], [6, 2, [0], Yes], [6, 3, [1], Yes], [6, 4, [5], Yes], [6, 5, [8], Yes], [6, 6, [9], Yes], [6, 7, [10], Yes], [6, 8, [11], Yes], [6, 9, [12], Yes], [6, 10, [2], Yes], [6, 11, [3], Yes], [6, 12, [4], Yes], [7, 0, [7], Yes], [7, 1, [8], Yes], [7, 2, [1], Yes], [7, 3, [5], Yes], [7, 4, [6], Yes], [7, 5, [9], Yes], [7, 6, [10], Yes], [7, 7, [11], Yes], [7, 8, [12], Yes], [7, 9, [2], Yes], [7, 10, [3], Yes], [7, 11, [4], Yes], [7, 12, [0], Yes], [8, 0, [8], Yes], [8, 1, [9], Yes], [8, 2, [5], Yes], [8, 3, [6], Yes], [8, 4, [7], Yes], [8, 5, [10], Yes], [8, 6, [11], Yes], [8, 7, [12], Yes], [8, 8, [2], Yes], [8, 9, [3], Yes], [8, 10, [4], Yes], [8, 11, [0], Yes], [8, 12, [1], Yes], [9, 0, [9], Yes], [9, 1, [10], Yes], [9, 2, [6], Yes], [9, 3, [7], Yes], [9, 4, [8], Yes], [9, 5, [11], Yes], [9, 6, [12], Yes], [9, 7, [2], Yes], [9, 8, [3], Yes], [9, 9, [4], Yes], [9, 10, [0], Yes], [9, 11, [1], Yes], [9, 12, [5], Yes], [10, 0, [10], Yes], [10, 1, [11], Yes], [10, 2, [7], Yes], [10, 3, [8], Yes], [10, 4, [9], Yes], [10, 5, [12], Yes], [10, 6, [2], Yes], [10, 7, [3], Yes], [10, 8, [4], Yes], [10, 9, [0], Yes], [10, 10, [1], Yes], [10, 11, [5], Yes], [10, 12, [6], Yes], [11, 0, [11], Yes], [11, 1, [12], Yes], [11, 2, [8], Yes], [11, 3, [9], Yes], [11, 4, [10], Yes], [11, 5, [2], Yes], [11, 6, [3], Yes], [11, 7, [4], Yes], [11, 8, [0], Yes], [11, 9, [1], Yes], [11, 10, [5], Yes], [11, 11, [6], Yes], [11, 12, [7], Yes], [12, 0, [12], Yes], [12, 1, [2], Yes], [12, 2, [9], Yes], [12, 3, [10], Yes], [12, 4, [11], Yes], [12, 5, [3], Yes], [12, 6, [4], Yes], [12, 7, [0], Yes], [12, 8, [1], Yes], [12, 9, [5], Yes], [12, 10, [6], Yes], [12, 11, [7], Yes], [12, 12, [8], Yes]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [10], [20], Yes], [3, [11], [22], Yes], [4, [12], [24], Yes], [5, [2], [4], Yes], [6, [3], [6], Yes], [7, [4], [8], Yes], [8, [5], [10], Yes], [9, [6], [12], Yes], [10, [7], [14], Yes], [11, [8], [16], Yes], [12, [9], [18], Yes]], Yes] Conditions: none Kernel normality gives(ah)(bk)=ab(b^-1hb)k in abK;check every representative product agrees on its quotient cell.Every f(g) in a coset must also agree before defining bar_f.The carrier consists of coset SETS,not least representatives. Step identifiers and dependenciesRule: Iso1_full_LEFT_coset_quotient_ALL_products_and_representative_images · Uses: G001, S008 S010 · impliesBefore source roles = [[[0], [1], [10], [11], [12], [2], [3], [4], [5], [6], [7], [8], [9]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], [0, 2, 20, 22, 24, 4, 6, 8, 10, 12, 14, 16, 18]] After derived = [[[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 20, 20, Yes], [0, 3, 22, 22, Yes], [0, 4, 24, 24, Yes], [0, 5, 4, 4, Yes], [0, 6, 6, 6, Yes], [0, 7, 8, 8, Yes], [0, 8, 10, 10, Yes], [0, 9, 12, 12, Yes], [0, 10, 14, 14, Yes], [0, 11, 16, 16, Yes], [0, 12, 18, 18, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 22, 22, Yes], [1, 3, 24, 24, Yes], [1, 4, 0, 0, Yes], [1, 5, 6, 6, Yes], [1, 6, 8, 8, Yes], [1, 7, 10, 10, Yes], [1, 8, 12, 12, Yes], [1, 9, 14, 14, Yes], [1, 10, 16, 16, Yes], [1, 11, 18, 18, Yes], [1, 12, 20, 20, Yes], [2, 0, 20, 20, Yes], [2, 1, 22, 22, Yes], [2, 2, 14, 14, Yes], [2, 3, 16, 16, Yes], [2, 4, 18, 18, Yes], [2, 5, 24, 24, Yes], [2, 6, 0, 0, Yes], [2, 7, 2, 2, Yes], [2, 8, 4, 4, Yes], [2, 9, 6, 6, Yes], [2, 10, 8, 8, Yes], [2, 11, 10, 10, Yes], [2, 12, 12, 12, Yes], [3, 0, 22, 22, Yes], [3, 1, 24, 24, Yes], [3, 2, 16, 16, Yes], [3, 3, 18, 18, Yes], [3, 4, 20, 20, Yes], [3, 5, 0, 0, Yes], [3, 6, 2, 2, Yes], [3, 7, 4, 4, Yes], [3, 8, 6, 6, Yes], [3, 9, 8, 8, Yes], [3, 10, 10, 10, Yes], [3, 11, 12, 12, Yes], [3, 12, 14, 14, Yes], [4, 0, 24, 24, Yes], [4, 1, 0, 0, Yes], [4, 2, 18, 18, Yes], [4, 3, 20, 20, Yes], [4, 4, 22, 22, Yes], [4, 5, 2, 2, Yes], [4, 6, 4, 4, Yes], [4, 7, 6, 6, Yes], [4, 8, 8, 8, Yes], [4, 9, 10, 10, Yes], [4, 10, 12, 12, Yes], [4, 11, 14, 14, Yes], [4, 12, 16, 16, Yes], [5, 0, 4, 4, Yes], [5, 1, 6, 6, Yes], [5, 2, 24, 24, Yes], [5, 3, 0, 0, Yes], [5, 4, 2, 2, Yes], [5, 5, 8, 8, Yes], [5, 6, 10, 10, Yes], [5, 7, 12, 12, Yes], [5, 8, 14, 14, Yes], [5, 9, 16, 16, Yes], [5, 10, 18, 18, Yes], [5, 11, 20, 20, Yes], [5, 12, 22, 22, Yes], [6, 0, 6, 6, Yes], [6, 1, 8, 8, Yes], [6, 2, 0, 0, Yes], [6, 3, 2, 2, Yes], [6, 4, 4, 4, Yes], [6, 5, 10, 10, Yes], [6, 6, 12, 12, Yes], [6, 7, 14, 14, Yes], [6, 8, 16, 16, Yes], [6, 9, 18, 18, Yes], [6, 10, 20, 20, Yes], [6, 11, 22, 22, Yes], [6, 12, 24, 24, Yes], [7, 0, 8, 8, Yes], [7, 1, 10, 10, Yes], [7, 2, 2, 2, Yes], [7, 3, 4, 4, Yes], [7, 4, 6, 6, Yes], [7, 5, 12, 12, Yes], [7, 6, 14, 14, Yes], [7, 7, 16, 16, Yes], [7, 8, 18, 18, Yes], [7, 9, 20, 20, Yes], [7, 10, 22, 22, Yes], [7, 11, 24, 24, Yes], [7, 12, 0, 0, Yes], [8, 0, 10, 10, Yes], [8, 1, 12, 12, Yes], [8, 2, 4, 4, Yes], [8, 3, 6, 6, Yes], [8, 4, 8, 8, Yes], [8, 5, 14, 14, Yes], [8, 6, 16, 16, Yes], [8, 7, 18, 18, Yes], [8, 8, 20, 20, Yes], [8, 9, 22, 22, Yes], [8, 10, 24, 24, Yes], [8, 11, 0, 0, Yes], [8, 12, 2, 2, Yes], [9, 0, 12, 12, Yes], [9, 1, 14, 14, Yes], [9, 2, 6, 6, Yes], [9, 3, 8, 8, Yes], [9, 4, 10, 10, Yes], [9, 5, 16, 16, Yes], [9, 6, 18, 18, Yes], [9, 7, 20, 20, Yes], [9, 8, 22, 22, Yes], [9, 9, 24, 24, Yes], [9, 10, 0, 0, Yes], [9, 11, 2, 2, Yes], [9, 12, 4, 4, Yes], [10, 0, 14, 14, Yes], [10, 1, 16, 16, Yes], [10, 2, 8, 8, Yes], [10, 3, 10, 10, Yes], [10, 4, 12, 12, Yes], [10, 5, 18, 18, Yes], [10, 6, 20, 20, Yes], [10, 7, 22, 22, Yes], [10, 8, 24, 24, Yes], [10, 9, 0, 0, Yes], [10, 10, 2, 2, Yes], [10, 11, 4, 4, Yes], [10, 12, 6, 6, Yes], [11, 0, 16, 16, Yes], [11, 1, 18, 18, Yes], [11, 2, 10, 10, Yes], [11, 3, 12, 12, Yes], [11, 4, 14, 14, Yes], [11, 5, 20, 20, Yes], [11, 6, 22, 22, Yes], [11, 7, 24, 24, Yes], [11, 8, 0, 0, Yes], [11, 9, 2, 2, Yes], [11, 10, 4, 4, Yes], [11, 11, 6, 6, Yes], [11, 12, 8, 8, Yes], [12, 0, 18, 18, Yes], [12, 1, 20, 20, Yes], [12, 2, 12, 12, Yes], [12, 3, 14, 14, Yes], [12, 4, 16, 16, Yes], [12, 5, 22, 22, Yes], [12, 6, 24, 24, Yes], [12, 7, 0, 0, Yes], [12, 8, 2, 2, Yes], [12, 9, 4, 4, Yes], [12, 10, 6, 6, Yes], [12, 11, 8, 8, Yes], [12, 12, 10, 10, Yes]], Yes, Yes, Yes, Yes, No] Conditions: none The identity f(ab)=f(a)f(b) preserves products.If f(a)=f(b),a^-1b is in K and the cosets agree,giving injectivity.Every image value f(g) is hit by gK,giving surjectivity onto the IMAGE.Equal cardinalities do not replace these obligations or give surjectivity to the whole target. Step identifiers and dependenciesRule: Iso1_ALL_induced_product_pairs_injective_and_onto_IMAGE_separate · Uses: G001, S009 S011 · impliesBefore source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13] After derived = [No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No] Conditions: none Only actual surjectivity identifies image with the whole codomain.The final No preserves general abstract/formal proof obligations,not denial of this complete finite isomorphism certificate or its pedagogical argument. Step identifiers and dependenciesRule: Iso1_image_VS_codomain_scope_and_finite_general_proof_boundary · Uses: G001, S010 ## Reverse checks - 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, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof) 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: 76261e7c994ac8973c9c2fa331f4c1495c528d1f90646b7a0abbef250e6c2b44 - candidate record SHA-256: dea80bcd8fdef82041c60ba011330ee17417de44af835914f5b0e29356c372d3 - semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82 - question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0087.json)