# Question 24945: Qualify Both Groups and Every Product Pair before Applying Kernel Theorems [Return to catalogue page 125](https://ocme.evemisslab.com/en/questions/page/125/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations/#q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0020) 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. R0048-24945 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0020 ## Original question and givens Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source 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], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], target identity=0, and total mapping=[0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate). - G001 — Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source 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], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], target identity=0, and total mapping=[0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, 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-group-homomorphism-kernel](https://ocme.evemisslab.com/en/mko/mko-finite-group-homomorphism-kernel/) ## Step-by-step derivation S001 · impliesBefore source roles = [kernel-homomorphism-qualification] After derived = [[source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate]] Conditions: none List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], 0] After derived = [22, [[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]], Yes] Conditions: none Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables. Step identifiers and dependenciesRule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001 S003 · impliesBefore source roles = [Kernel ORIGINAL SOURCE group, 22, FOR EVERY x,y,z in0..n-1] 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]], 10648, Yes] Conditions: none Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups. Step identifiers and dependenciesRule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002 S004 · impliesBefore source roles = [Kernel ORIGINAL SOURCE group, 0] After derived = [[0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 21, 0, 0], [2, 20, 0, 0], [3, 19, 0, 0], [4, 18, 0, 0], [5, 17, 0, 0], [6, 16, 0, 0], [7, 15, 0, 0], [8, 14, 0, 0], [9, 13, 0, 0], [10, 12, 0, 0], [11, 11, 0, 0], [12, 10, 0, 0], [13, 9, 0, 0], [14, 8, 0, 0], [15, 7, 0, 0], [16, 6, 0, 0], [17, 5, 0, 0], [18, 4, 0, 0], [19, 3, 0, 0], [20, 2, 0, 0], [21, 1, 0, 0]], Yes] Conditions: none Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises. Step identifiers and dependenciesRule: Group_ALL_two_sided_inverses · Uses: G001, S003 S005 · impliesBefore source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], 0] After derived = [22, [[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]], Yes] Conditions: none Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables. Step identifiers and dependenciesRule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S004 S006 · impliesBefore source roles = [Kernel ORIGINAL TARGET group, 22, FOR EVERY x,y,z in0..n-1] 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]], 10648, Yes] Conditions: none Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups. Step identifiers and dependenciesRule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S005 S007 · impliesBefore source roles = [Kernel ORIGINAL TARGET group, 0] After derived = [[0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 21, 0, 0], [2, 20, 0, 0], [3, 19, 0, 0], [4, 18, 0, 0], [5, 17, 0, 0], [6, 16, 0, 0], [7, 15, 0, 0], [8, 14, 0, 0], [9, 13, 0, 0], [10, 12, 0, 0], [11, 11, 0, 0], [12, 10, 0, 0], [13, 9, 0, 0], [14, 8, 0, 0], [15, 7, 0, 0], [16, 6, 0, 0], [17, 5, 0, 0], [18, 4, 0, 0], [19, 3, 0, 0], [20, 2, 0, 0], [21, 1, 0, 0]], Yes] Conditions: none Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises. Step identifiers and dependenciesRule: Group_ALL_two_sided_inverses · Uses: G001, S006 S008 · impliesBefore source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]] After derived = [[[0, 0, 0, 0, Yes], [0, 1, 21, 21, Yes], [0, 2, 20, 20, Yes], [0, 3, 19, 19, Yes], [0, 4, 18, 18, Yes], [0, 5, 17, 17, Yes], [0, 6, 16, 16, Yes], [0, 7, 15, 15, Yes], [0, 8, 14, 14, Yes], [0, 9, 13, 13, Yes], [0, 10, 12, 12, Yes], [0, 11, 11, 11, Yes], [0, 12, 10, 10, Yes], [0, 13, 9, 9, Yes], [0, 14, 8, 8, Yes], [0, 15, 7, 7, Yes], [0, 16, 6, 6, Yes], [0, 17, 5, 5, Yes], [0, 18, 4, 4, Yes], [0, 19, 3, 3, Yes], [0, 20, 2, 2, Yes], [0, 21, 1, 1, Yes], [1, 0, 21, 21, Yes], [1, 1, 20, 20, Yes], [1, 2, 19, 19, Yes], [1, 3, 18, 18, Yes], [1, 4, 17, 17, Yes], [1, 5, 16, 16, Yes], [1, 6, 15, 15, Yes], [1, 7, 14, 14, Yes], [1, 8, 13, 13, Yes], [1, 9, 12, 12, Yes], [1, 10, 11, 11, Yes], [1, 11, 10, 10, Yes], [1, 12, 9, 9, Yes], [1, 13, 8, 8, Yes], [1, 14, 7, 7, Yes], [1, 15, 6, 6, Yes], [1, 16, 5, 5, Yes], [1, 17, 4, 4, Yes], [1, 18, 3, 3, Yes], [1, 19, 2, 2, Yes], [1, 20, 1, 1, Yes], [1, 21, 0, 0, Yes], [2, 0, 20, 20, Yes], [2, 1, 19, 19, Yes], [2, 2, 18, 18, Yes], [2, 3, 17, 17, Yes], [2, 4, 16, 16, Yes], [2, 5, 15, 15, Yes], [2, 6, 14, 14, Yes], [2, 7, 13, 13, Yes], [2, 8, 12, 12, Yes], [2, 9, 11, 11, Yes], [2, 10, 10, 10, Yes], [2, 11, 9, 9, Yes], [2, 12, 8, 8, Yes], [2, 13, 7, 7, Yes], [2, 14, 6, 6, Yes], [2, 15, 5, 5, Yes], [2, 16, 4, 4, Yes], [2, 17, 3, 3, Yes], [2, 18, 2, 2, Yes], [2, 19, 1, 1, Yes], [2, 20, 0, 0, Yes], [2, 21, 21, 21, Yes], [3, 0, 19, 19, Yes], [3, 1, 18, 18, Yes], [3, 2, 17, 17, Yes], [3, 3, 16, 16, Yes], [3, 4, 15, 15, Yes], [3, 5, 14, 14, Yes], [3, 6, 13, 13, Yes], [3, 7, 12, 12, Yes], [3, 8, 11, 11, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [3, 11, 8, 8, Yes], [3, 12, 7, 7, Yes], [3, 13, 6, 6, Yes], [3, 14, 5, 5, Yes], [3, 15, 4, 4, Yes], [3, 16, 3, 3, Yes], [3, 17, 2, 2, Yes], [3, 18, 1, 1, Yes], [3, 19, 0, 0, Yes], [3, 20, 21, 21, Yes], [3, 21, 20, 20, Yes], [4, 0, 18, 18, Yes], [4, 1, 17, 17, Yes], [4, 2, 16, 16, Yes], [4, 3, 15, 15, Yes], [4, 4, 14, 14, Yes], [4, 5, 13, 13, Yes], [4, 6, 12, 12, Yes], [4, 7, 11, 11, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [4, 11, 7, 7, Yes], [4, 12, 6, 6, Yes], [4, 13, 5, 5, Yes], [4, 14, 4, 4, Yes], [4, 15, 3, 3, Yes], [4, 16, 2, 2, Yes], [4, 17, 1, 1, Yes], [4, 18, 0, 0, Yes], [4, 19, 21, 21, Yes], [4, 20, 20, 20, Yes], [4, 21, 19, 19, Yes], [5, 0, 17, 17, Yes], [5, 1, 16, 16, Yes], [5, 2, 15, 15, Yes], [5, 3, 14, 14, Yes], [5, 4, 13, 13, Yes], [5, 5, 12, 12, Yes], [5, 6, 11, 11, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [5, 11, 6, 6, Yes], [5, 12, 5, 5, Yes], [5, 13, 4, 4, Yes], [5, 14, 3, 3, Yes], [5, 15, 2, 2, Yes], [5, 16, 1, 1, Yes], [5, 17, 0, 0, Yes], [5, 18, 21, 21, Yes], [5, 19, 20, 20, Yes], [5, 20, 19, 19, Yes], [5, 21, 18, 18, Yes], [6, 0, 16, 16, Yes], [6, 1, 15, 15, Yes], [6, 2, 14, 14, Yes], [6, 3, 13, 13, Yes], [6, 4, 12, 12, Yes], [6, 5, 11, 11, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [6, 11, 5, 5, Yes], [6, 12, 4, 4, Yes], [6, 13, 3, 3, Yes], [6, 14, 2, 2, Yes], [6, 15, 1, 1, Yes], [6, 16, 0, 0, Yes], [6, 17, 21, 21, Yes], [6, 18, 20, 20, Yes], [6, 19, 19, 19, Yes], [6, 20, 18, 18, Yes], [6, 21, 17, 17, Yes], [7, 0, 15, 15, Yes], [7, 1, 14, 14, Yes], [7, 2, 13, 13, Yes], [7, 3, 12, 12, Yes], [7, 4, 11, 11, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [7, 11, 4, 4, Yes], [7, 12, 3, 3, Yes], [7, 13, 2, 2, Yes], [7, 14, 1, 1, Yes], [7, 15, 0, 0, Yes], [7, 16, 21, 21, Yes], [7, 17, 20, 20, Yes], [7, 18, 19, 19, Yes], [7, 19, 18, 18, Yes], [7, 20, 17, 17, Yes], [7, 21, 16, 16, Yes], [8, 0, 14, 14, Yes], [8, 1, 13, 13, Yes], [8, 2, 12, 12, Yes], [8, 3, 11, 11, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [8, 11, 3, 3, Yes], [8, 12, 2, 2, Yes], [8, 13, 1, 1, Yes], [8, 14, 0, 0, Yes], [8, 15, 21, 21, Yes], [8, 16, 20, 20, Yes], [8, 17, 19, 19, Yes], [8, 18, 18, 18, Yes], [8, 19, 17, 17, Yes], [8, 20, 16, 16, Yes], [8, 21, 15, 15, Yes], [9, 0, 13, 13, Yes], [9, 1, 12, 12, Yes], [9, 2, 11, 11, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [9, 11, 2, 2, Yes], [9, 12, 1, 1, Yes], [9, 13, 0, 0, Yes], [9, 14, 21, 21, Yes], [9, 15, 20, 20, Yes], [9, 16, 19, 19, Yes], [9, 17, 18, 18, Yes], [9, 18, 17, 17, Yes], [9, 19, 16, 16, Yes], [9, 20, 15, 15, Yes], [9, 21, 14, 14, Yes], [10, 0, 12, 12, Yes], [10, 1, 11, 11, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes], [10, 11, 1, 1, Yes], [10, 12, 0, 0, Yes], [10, 13, 21, 21, Yes], [10, 14, 20, 20, Yes], [10, 15, 19, 19, Yes], [10, 16, 18, 18, Yes], [10, 17, 17, 17, Yes], [10, 18, 16, 16, Yes], [10, 19, 15, 15, Yes], [10, 20, 14, 14, Yes], [10, 21, 13, 13, Yes], [11, 0, 11, 11, Yes], [11, 1, 10, 10, Yes], [11, 2, 9, 9, Yes], [11, 3, 8, 8, Yes], [11, 4, 7, 7, Yes], [11, 5, 6, 6, Yes], [11, 6, 5, 5, Yes], [11, 7, 4, 4, Yes], [11, 8, 3, 3, Yes], [11, 9, 2, 2, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, Yes], [11, 12, 21, 21, Yes], [11, 13, 20, 20, Yes], [11, 14, 19, 19, Yes], [11, 15, 18, 18, Yes], [11, 16, 17, 17, Yes], [11, 17, 16, 16, Yes], [11, 18, 15, 15, Yes], [11, 19, 14, 14, Yes], [11, 20, 13, 13, Yes], [11, 21, 12, 12, Yes], [12, 0, 10, 10, Yes], [12, 1, 9, 9, Yes], [12, 2, 8, 8, Yes], [12, 3, 7, 7, Yes], [12, 4, 6, 6, Yes], [12, 5, 5, 5, Yes], [12, 6, 4, 4, Yes], [12, 7, 3, 3, Yes], [12, 8, 2, 2, Yes], [12, 9, 1, 1, Yes], [12, 10, 0, 0, Yes], [12, 11, 21, 21, Yes], [12, 12, 20, 20, Yes], [12, 13, 19, 19, Yes], [12, 14, 18, 18, Yes], [12, 15, 17, 17, Yes], [12, 16, 16, 16, Yes], [12, 17, 15, 15, Yes], [12, 18, 14, 14, Yes], [12, 19, 13, 13, Yes], [12, 20, 12, 12, Yes], [12, 21, 11, 11, Yes], [13, 0, 9, 9, Yes], [13, 1, 8, 8, Yes], [13, 2, 7, 7, Yes], [13, 3, 6, 6, Yes], [13, 4, 5, 5, Yes], [13, 5, 4, 4, Yes], [13, 6, 3, 3, Yes], [13, 7, 2, 2, Yes], [13, 8, 1, 1, Yes], [13, 9, 0, 0, Yes], [13, 10, 21, 21, Yes], [13, 11, 20, 20, Yes], [13, 12, 19, 19, Yes], [13, 13, 18, 18, Yes], [13, 14, 17, 17, Yes], [13, 15, 16, 16, Yes], [13, 16, 15, 15, Yes], [13, 17, 14, 14, Yes], [13, 18, 13, 13, Yes], [13, 19, 12, 12, Yes], [13, 20, 11, 11, Yes], [13, 21, 10, 10, Yes], [14, 0, 8, 8, Yes], [14, 1, 7, 7, Yes], [14, 2, 6, 6, Yes], [14, 3, 5, 5, Yes], [14, 4, 4, 4, Yes], [14, 5, 3, 3, Yes], [14, 6, 2, 2, Yes], [14, 7, 1, 1, Yes], [14, 8, 0, 0, Yes], [14, 9, 21, 21, Yes], [14, 10, 20, 20, Yes], [14, 11, 19, 19, Yes], [14, 12, 18, 18, Yes], [14, 13, 17, 17, Yes], [14, 14, 16, 16, Yes], [14, 15, 15, 15, Yes], [14, 16, 14, 14, Yes], [14, 17, 13, 13, Yes], [14, 18, 12, 12, Yes], [14, 19, 11, 11, Yes], [14, 20, 10, 10, Yes], [14, 21, 9, 9, Yes], [15, 0, 7, 7, Yes], [15, 1, 6, 6, Yes], [15, 2, 5, 5, Yes], [15, 3, 4, 4, Yes], [15, 4, 3, 3, Yes], [15, 5, 2, 2, Yes], [15, 6, 1, 1, Yes], [15, 7, 0, 0, Yes], [15, 8, 21, 21, Yes], [15, 9, 20, 20, Yes], [15, 10, 19, 19, Yes], [15, 11, 18, 18, Yes], [15, 12, 17, 17, Yes], [15, 13, 16, 16, Yes], [15, 14, 15, 15, Yes], [15, 15, 14, 14, Yes], [15, 16, 13, 13, Yes], [15, 17, 12, 12, Yes], [15, 18, 11, 11, Yes], [15, 19, 10, 10, Yes], [15, 20, 9, 9, Yes], [15, 21, 8, 8, Yes], [16, 0, 6, 6, Yes], [16, 1, 5, 5, Yes], [16, 2, 4, 4, Yes], [16, 3, 3, 3, Yes], [16, 4, 2, 2, Yes], [16, 5, 1, 1, Yes], [16, 6, 0, 0, Yes], [16, 7, 21, 21, Yes], [16, 8, 20, 20, Yes], [16, 9, 19, 19, Yes], [16, 10, 18, 18, Yes], [16, 11, 17, 17, Yes], [16, 12, 16, 16, Yes], [16, 13, 15, 15, Yes], [16, 14, 14, 14, Yes], [16, 15, 13, 13, Yes], [16, 16, 12, 12, Yes], [16, 17, 11, 11, Yes], [16, 18, 10, 10, Yes], [16, 19, 9, 9, Yes], [16, 20, 8, 8, Yes], [16, 21, 7, 7, Yes], [17, 0, 5, 5, Yes], [17, 1, 4, 4, Yes], [17, 2, 3, 3, Yes], [17, 3, 2, 2, Yes], [17, 4, 1, 1, Yes], [17, 5, 0, 0, Yes], [17, 6, 21, 21, Yes], [17, 7, 20, 20, Yes], [17, 8, 19, 19, Yes], [17, 9, 18, 18, Yes], [17, 10, 17, 17, Yes], [17, 11, 16, 16, Yes], [17, 12, 15, 15, Yes], [17, 13, 14, 14, Yes], [17, 14, 13, 13, Yes], [17, 15, 12, 12, Yes], [17, 16, 11, 11, Yes], [17, 17, 10, 10, Yes], [17, 18, 9, 9, Yes], [17, 19, 8, 8, Yes], [17, 20, 7, 7, Yes], [17, 21, 6, 6, Yes], [18, 0, 4, 4, Yes], [18, 1, 3, 3, Yes], [18, 2, 2, 2, Yes], [18, 3, 1, 1, Yes], [18, 4, 0, 0, Yes], [18, 5, 21, 21, Yes], [18, 6, 20, 20, Yes], [18, 7, 19, 19, Yes], [18, 8, 18, 18, Yes], [18, 9, 17, 17, Yes], [18, 10, 16, 16, Yes], [18, 11, 15, 15, Yes], [18, 12, 14, 14, Yes], [18, 13, 13, 13, Yes], [18, 14, 12, 12, Yes], [18, 15, 11, 11, Yes], [18, 16, 10, 10, Yes], [18, 17, 9, 9, Yes], [18, 18, 8, 8, Yes], [18, 19, 7, 7, Yes], [18, 20, 6, 6, Yes], [18, 21, 5, 5, Yes], [19, 0, 3, 3, Yes], [19, 1, 2, 2, Yes], [19, 2, 1, 1, Yes], [19, 3, 0, 0, Yes], [19, 4, 21, 21, Yes], [19, 5, 20, 20, Yes], [19, 6, 19, 19, Yes], [19, 7, 18, 18, Yes], [19, 8, 17, 17, Yes], [19, 9, 16, 16, Yes], [19, 10, 15, 15, Yes], [19, 11, 14, 14, Yes], [19, 12, 13, 13, Yes], [19, 13, 12, 12, Yes], [19, 14, 11, 11, Yes], [19, 15, 10, 10, Yes], [19, 16, 9, 9, Yes], [19, 17, 8, 8, Yes], [19, 18, 7, 7, Yes], [19, 19, 6, 6, Yes], [19, 20, 5, 5, Yes], [19, 21, 4, 4, Yes], [20, 0, 2, 2, Yes], [20, 1, 1, 1, Yes], [20, 2, 0, 0, Yes], [20, 3, 21, 21, Yes], [20, 4, 20, 20, Yes], [20, 5, 19, 19, Yes], [20, 6, 18, 18, Yes], [20, 7, 17, 17, Yes], [20, 8, 16, 16, Yes], [20, 9, 15, 15, Yes], [20, 10, 14, 14, Yes], [20, 11, 13, 13, Yes], [20, 12, 12, 12, Yes], [20, 13, 11, 11, Yes], [20, 14, 10, 10, Yes], [20, 15, 9, 9, Yes], [20, 16, 8, 8, Yes], [20, 17, 7, 7, Yes], [20, 18, 6, 6, Yes], [20, 19, 5, 5, Yes], [20, 20, 4, 4, Yes], [20, 21, 3, 3, Yes], [21, 0, 1, 1, Yes], [21, 1, 0, 0, Yes], [21, 2, 21, 21, Yes], [21, 3, 20, 20, Yes], [21, 4, 19, 19, Yes], [21, 5, 18, 18, Yes], [21, 6, 17, 17, Yes], [21, 7, 16, 16, Yes], [21, 8, 15, 15, Yes], [21, 9, 14, 14, Yes], [21, 10, 13, 13, Yes], [21, 11, 12, 12, Yes], [21, 12, 11, 11, Yes], [21, 13, 10, 10, Yes], [21, 14, 9, 9, Yes], [21, 15, 8, 8, Yes], [21, 16, 7, 7, Yes], [21, 17, 6, 6, Yes], [21, 18, 5, 5, Yes], [21, 19, 4, 4, Yes], [21, 20, 3, 3, Yes], [21, 21, 2, 2, Yes]], Yes, null] Conditions: none For these same source and target groups,check f(xy)=f(x)f(y) for every pair and retain the first failed row/null.A total map need not be a homomorphism.Failure blocks kernel theorems;an internal fallback is not the true identity preimage. Step identifiers and dependenciesRule: Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems · Uses: G001, S007 ## Reverse checks - 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 (22, 22, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 21, 21, Yes], [0, 2, 20, 20, Yes], [0, 3, 19, 19, Yes], [0, 4, 18, 18, Yes], [0, 5, 17, 17, Yes], [0, 6, 16, 16, Yes], [0, 7, 15, 15, Yes], [0, 8, 14, 14, Yes], [0, 9, 13, 13, Yes], [0, 10, 12, 12, Yes], [0, 11, 11, 11, Yes], [0, 12, 10, 10, Yes], [0, 13, 9, 9, Yes], [0, 14, 8, 8, Yes], [0, 15, 7, 7, Yes], [0, 16, 6, 6, Yes], [0, 17, 5, 5, Yes], [0, 18, 4, 4, Yes], [0, 19, 3, 3, Yes], [0, 20, 2, 2, Yes], [0, 21, 1, 1, Yes], [1, 0, 21, 21, Yes], [1, 1, 20, 20, Yes], [1, 2, 19, 19, Yes], [1, 3, 18, 18, Yes], [1, 4, 17, 17, Yes], [1, 5, 16, 16, Yes], [1, 6, 15, 15, Yes], [1, 7, 14, 14, Yes], [1, 8, 13, 13, Yes], [1, 9, 12, 12, Yes], [1, 10, 11, 11, Yes], [1, 11, 10, 10, Yes], [1, 12, 9, 9, Yes], [1, 13, 8, 8, Yes], [1, 14, 7, 7, Yes], [1, 15, 6, 6, Yes], [1, 16, 5, 5, Yes], [1, 17, 4, 4, Yes], [1, 18, 3, 3, Yes], [1, 19, 2, 2, Yes], [1, 20, 1, 1, Yes], [1, 21, 0, 0, Yes], [2, 0, 20, 20, Yes], [2, 1, 19, 19, Yes], [2, 2, 18, 18, Yes], [2, 3, 17, 17, Yes], [2, 4, 16, 16, Yes], [2, 5, 15, 15, Yes], [2, 6, 14, 14, Yes], [2, 7, 13, 13, Yes], [2, 8, 12, 12, Yes], [2, 9, 11, 11, Yes], [2, 10, 10, 10, Yes], [2, 11, 9, 9, Yes], [2, 12, 8, 8, Yes], [2, 13, 7, 7, Yes], [2, 14, 6, 6, Yes], [2, 15, 5, 5, Yes], [2, 16, 4, 4, Yes], [2, 17, 3, 3, Yes], [2, 18, 2, 2, Yes], [2, 19, 1, 1, Yes], [2, 20, 0, 0, Yes], [2, 21, 21, 21, Yes], [3, 0, 19, 19, Yes], [3, 1, 18, 18, Yes], [3, 2, 17, 17, Yes], [3, 3, 16, 16, Yes], [3, 4, 15, 15, Yes], [3, 5, 14, 14, Yes], [3, 6, 13, 13, Yes], [3, 7, 12, 12, Yes], [3, 8, 11, 11, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [3, 11, 8, 8, Yes], [3, 12, 7, 7, Yes], [3, 13, 6, 6, Yes], [3, 14, 5, 5, Yes], [3, 15, 4, 4, Yes], [3, 16, 3, 3, Yes], [3, 17, 2, 2, Yes], [3, 18, 1, 1, Yes], [3, 19, 0, 0, Yes], [3, 20, 21, 21, Yes], [3, 21, 20, 20, Yes], [4, 0, 18, 18, Yes], [4, 1, 17, 17, Yes], [4, 2, 16, 16, Yes], [4, 3, 15, 15, Yes], [4, 4, 14, 14, Yes], [4, 5, 13, 13, Yes], [4, 6, 12, 12, Yes], [4, 7, 11, 11, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [4, 11, 7, 7, Yes], [4, 12, 6, 6, Yes], [4, 13, 5, 5, Yes], [4, 14, 4, 4, Yes], [4, 15, 3, 3, Yes], [4, 16, 2, 2, Yes], [4, 17, 1, 1, Yes], [4, 18, 0, 0, Yes], [4, 19, 21, 21, Yes], [4, 20, 20, 20, Yes], [4, 21, 19, 19, Yes], [5, 0, 17, 17, Yes], [5, 1, 16, 16, Yes], [5, 2, 15, 15, Yes], [5, 3, 14, 14, Yes], [5, 4, 13, 13, Yes], [5, 5, 12, 12, Yes], [5, 6, 11, 11, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [5, 11, 6, 6, Yes], [5, 12, 5, 5, Yes], [5, 13, 4, 4, Yes], [5, 14, 3, 3, Yes], [5, 15, 2, 2, Yes], [5, 16, 1, 1, Yes], [5, 17, 0, 0, Yes], [5, 18, 21, 21, Yes], [5, 19, 20, 20, Yes], [5, 20, 19, 19, Yes], [5, 21, 18, 18, Yes], [6, 0, 16, 16, Yes], [6, 1, 15, 15, Yes], [6, 2, 14, 14, Yes], [6, 3, 13, 13, Yes], [6, 4, 12, 12, Yes], [6, 5, 11, 11, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [6, 11, 5, 5, Yes], [6, 12, 4, 4, Yes], [6, 13, 3, 3, Yes], [6, 14, 2, 2, Yes], [6, 15, 1, 1, Yes], [6, 16, 0, 0, Yes], [6, 17, 21, 21, Yes], [6, 18, 20, 20, Yes], [6, 19, 19, 19, Yes], [6, 20, 18, 18, Yes], [6, 21, 17, 17, Yes], [7, 0, 15, 15, Yes], [7, 1, 14, 14, Yes], [7, 2, 13, 13, Yes], [7, 3, 12, 12, Yes], [7, 4, 11, 11, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [7, 11, 4, 4, Yes], [7, 12, 3, 3, Yes], [7, 13, 2, 2, Yes], [7, 14, 1, 1, Yes], [7, 15, 0, 0, Yes], [7, 16, 21, 21, Yes], [7, 17, 20, 20, Yes], [7, 18, 19, 19, Yes], [7, 19, 18, 18, Yes], [7, 20, 17, 17, Yes], [7, 21, 16, 16, Yes], [8, 0, 14, 14, Yes], [8, 1, 13, 13, Yes], [8, 2, 12, 12, Yes], [8, 3, 11, 11, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [8, 11, 3, 3, Yes], [8, 12, 2, 2, Yes], [8, 13, 1, 1, Yes], [8, 14, 0, 0, Yes], [8, 15, 21, 21, Yes], [8, 16, 20, 20, Yes], [8, 17, 19, 19, Yes], [8, 18, 18, 18, Yes], [8, 19, 17, 17, Yes], [8, 20, 16, 16, Yes], [8, 21, 15, 15, Yes], [9, 0, 13, 13, Yes], [9, 1, 12, 12, Yes], [9, 2, 11, 11, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [9, 11, 2, 2, Yes], [9, 12, 1, 1, Yes], [9, 13, 0, 0, Yes], [9, 14, 21, 21, Yes], [9, 15, 20, 20, Yes], [9, 16, 19, 19, Yes], [9, 17, 18, 18, Yes], [9, 18, 17, 17, Yes], [9, 19, 16, 16, Yes], [9, 20, 15, 15, Yes], [9, 21, 14, 14, Yes], [10, 0, 12, 12, Yes], [10, 1, 11, 11, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes], [10, 11, 1, 1, Yes], [10, 12, 0, 0, Yes], [10, 13, 21, 21, Yes], [10, 14, 20, 20, Yes], [10, 15, 19, 19, Yes], [10, 16, 18, 18, Yes], [10, 17, 17, 17, Yes], [10, 18, 16, 16, Yes], [10, 19, 15, 15, Yes], [10, 20, 14, 14, Yes], [10, 21, 13, 13, Yes], [11, 0, 11, 11, Yes], [11, 1, 10, 10, Yes], [11, 2, 9, 9, Yes], [11, 3, 8, 8, Yes], [11, 4, 7, 7, Yes], [11, 5, 6, 6, Yes], [11, 6, 5, 5, Yes], [11, 7, 4, 4, Yes], [11, 8, 3, 3, Yes], [11, 9, 2, 2, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, Yes], [11, 12, 21, 21, Yes], [11, 13, 20, 20, Yes], [11, 14, 19, 19, Yes], [11, 15, 18, 18, Yes], [11, 16, 17, 17, Yes], [11, 17, 16, 16, Yes], [11, 18, 15, 15, Yes], [11, 19, 14, 14, Yes], [11, 20, 13, 13, Yes], [11, 21, 12, 12, Yes], [12, 0, 10, 10, Yes], [12, 1, 9, 9, Yes], [12, 2, 8, 8, Yes], [12, 3, 7, 7, Yes], [12, 4, 6, 6, Yes], [12, 5, 5, 5, Yes], [12, 6, 4, 4, Yes], [12, 7, 3, 3, Yes], [12, 8, 2, 2, Yes], [12, 9, 1, 1, Yes], [12, 10, 0, 0, Yes], [12, 11, 21, 21, Yes], [12, 12, 20, 20, Yes], [12, 13, 19, 19, Yes], [12, 14, 18, 18, Yes], [12, 15, 17, 17, Yes], [12, 16, 16, 16, Yes], [12, 17, 15, 15, Yes], [12, 18, 14, 14, Yes], [12, 19, 13, 13, Yes], [12, 20, 12, 12, Yes], [12, 21, 11, 11, Yes], [13, 0, 9, 9, Yes], [13, 1, 8, 8, Yes], [13, 2, 7, 7, Yes], [13, 3, 6, 6, Yes], [13, 4, 5, 5, Yes], [13, 5, 4, 4, Yes], [13, 6, 3, 3, Yes], [13, 7, 2, 2, Yes], [13, 8, 1, 1, Yes], [13, 9, 0, 0, Yes], [13, 10, 21, 21, Yes], [13, 11, 20, 20, Yes], [13, 12, 19, 19, Yes], [13, 13, 18, 18, Yes], [13, 14, 17, 17, Yes], [13, 15, 16, 16, Yes], [13, 16, 15, 15, Yes], [13, 17, 14, 14, Yes], [13, 18, 13, 13, Yes], [13, 19, 12, 12, Yes], [13, 20, 11, 11, Yes], [13, 21, 10, 10, Yes], [14, 0, 8, 8, Yes], [14, 1, 7, 7, Yes], [14, 2, 6, 6, Yes], [14, 3, 5, 5, Yes], [14, 4, 4, 4, Yes], [14, 5, 3, 3, Yes], [14, 6, 2, 2, Yes], [14, 7, 1, 1, Yes], [14, 8, 0, 0, Yes], [14, 9, 21, 21, Yes], [14, 10, 20, 20, Yes], [14, 11, 19, 19, Yes], [14, 12, 18, 18, Yes], [14, 13, 17, 17, Yes], [14, 14, 16, 16, Yes], [14, 15, 15, 15, Yes], [14, 16, 14, 14, Yes], [14, 17, 13, 13, Yes], [14, 18, 12, 12, Yes], [14, 19, 11, 11, Yes], [14, 20, 10, 10, Yes], [14, 21, 9, 9, Yes], [15, 0, 7, 7, Yes], [15, 1, 6, 6, Yes], [15, 2, 5, 5, Yes], [15, 3, 4, 4, Yes], [15, 4, 3, 3, Yes], [15, 5, 2, 2, Yes], [15, 6, 1, 1, Yes], [15, 7, 0, 0, Yes], [15, 8, 21, 21, Yes], [15, 9, 20, 20, Yes], [15, 10, 19, 19, Yes], [15, 11, 18, 18, Yes], [15, 12, 17, 17, Yes], [15, 13, 16, 16, Yes], [15, 14, 15, 15, Yes], [15, 15, 14, 14, Yes], [15, 16, 13, 13, Yes], [15, 17, 12, 12, Yes], [15, 18, 11, 11, Yes], [15, 19, 10, 10, Yes], [15, 20, 9, 9, Yes], [15, 21, 8, 8, Yes], [16, 0, 6, 6, Yes], [16, 1, 5, 5, Yes], [16, 2, 4, 4, Yes], [16, 3, 3, 3, Yes], [16, 4, 2, 2, Yes], [16, 5, 1, 1, Yes], [16, 6, 0, 0, Yes], [16, 7, 21, 21, Yes], [16, 8, 20, 20, Yes], [16, 9, 19, 19, Yes], [16, 10, 18, 18, Yes], [16, 11, 17, 17, Yes], [16, 12, 16, 16, Yes], [16, 13, 15, 15, Yes], [16, 14, 14, 14, Yes], [16, 15, 13, 13, Yes], [16, 16, 12, 12, Yes], [16, 17, 11, 11, Yes], [16, 18, 10, 10, Yes], [16, 19, 9, 9, Yes], [16, 20, 8, 8, Yes], [16, 21, 7, 7, Yes], [17, 0, 5, 5, Yes], [17, 1, 4, 4, Yes], [17, 2, 3, 3, Yes], [17, 3, 2, 2, Yes], [17, 4, 1, 1, Yes], [17, 5, 0, 0, Yes], [17, 6, 21, 21, Yes], [17, 7, 20, 20, Yes], [17, 8, 19, 19, Yes], [17, 9, 18, 18, Yes], [17, 10, 17, 17, Yes], [17, 11, 16, 16, Yes], [17, 12, 15, 15, Yes], [17, 13, 14, 14, Yes], [17, 14, 13, 13, Yes], [17, 15, 12, 12, Yes], [17, 16, 11, 11, Yes], [17, 17, 10, 10, Yes], [17, 18, 9, 9, Yes], [17, 19, 8, 8, Yes], [17, 20, 7, 7, Yes], [17, 21, 6, 6, Yes], [18, 0, 4, 4, Yes], [18, 1, 3, 3, Yes], [18, 2, 2, 2, Yes], [18, 3, 1, 1, Yes], [18, 4, 0, 0, Yes], [18, 5, 21, 21, Yes], [18, 6, 20, 20, Yes], [18, 7, 19, 19, Yes], [18, 8, 18, 18, Yes], [18, 9, 17, 17, Yes], [18, 10, 16, 16, Yes], [18, 11, 15, 15, Yes], [18, 12, 14, 14, Yes], [18, 13, 13, 13, Yes], [18, 14, 12, 12, Yes], [18, 15, 11, 11, Yes], [18, 16, 10, 10, Yes], [18, 17, 9, 9, Yes], [18, 18, 8, 8, Yes], [18, 19, 7, 7, Yes], [18, 20, 6, 6, Yes], [18, 21, 5, 5, Yes], [19, 0, 3, 3, Yes], [19, 1, 2, 2, Yes], [19, 2, 1, 1, Yes], [19, 3, 0, 0, Yes], [19, 4, 21, 21, Yes], [19, 5, 20, 20, Yes], [19, 6, 19, 19, Yes], [19, 7, 18, 18, Yes], [19, 8, 17, 17, Yes], [19, 9, 16, 16, Yes], [19, 10, 15, 15, Yes], [19, 11, 14, 14, Yes], [19, 12, 13, 13, Yes], [19, 13, 12, 12, Yes], [19, 14, 11, 11, Yes], [19, 15, 10, 10, Yes], [19, 16, 9, 9, Yes], [19, 17, 8, 8, Yes], [19, 18, 7, 7, Yes], [19, 19, 6, 6, Yes], [19, 20, 5, 5, Yes], [19, 21, 4, 4, Yes], [20, 0, 2, 2, Yes], [20, 1, 1, 1, Yes], [20, 2, 0, 0, Yes], [20, 3, 21, 21, Yes], [20, 4, 20, 20, Yes], [20, 5, 19, 19, Yes], [20, 6, 18, 18, Yes], [20, 7, 17, 17, Yes], [20, 8, 16, 16, Yes], [20, 9, 15, 15, Yes], [20, 10, 14, 14, Yes], [20, 11, 13, 13, Yes], [20, 12, 12, 12, Yes], [20, 13, 11, 11, Yes], [20, 14, 10, 10, Yes], [20, 15, 9, 9, Yes], [20, 16, 8, 8, Yes], [20, 17, 7, 7, Yes], [20, 18, 6, 6, Yes], [20, 19, 5, 5, Yes], [20, 20, 4, 4, Yes], [20, 21, 3, 3, Yes], [21, 0, 1, 1, Yes], [21, 1, 0, 0, Yes], [21, 2, 21, 21, Yes], [21, 3, 20, 20, Yes], [21, 4, 19, 19, Yes], [21, 5, 18, 18, Yes], [21, 6, 17, 17, Yes], [21, 7, 16, 16, Yes], [21, 8, 15, 15, Yes], [21, 9, 14, 14, Yes], [21, 10, 13, 13, Yes], [21, 11, 12, 12, Yes], [21, 12, 11, 11, Yes], [21, 13, 10, 10, Yes], [21, 14, 9, 9, Yes], [21, 15, 8, 8, Yes], [21, 16, 7, 7, Yes], [21, 17, 6, 6, Yes], [21, 18, 5, 5, Yes], [21, 19, 4, 4, Yes], [21, 20, 3, 3, Yes], [21, 21, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair) 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: 7d82a75149208e68e0925e1bc617b3aa8163f4b60f44c3bdbf323e0ce832a63c - candidate record SHA-256: 1d8553a96317fa470111a7f8e50f63fb0a35e1857edc147f5258b02baa79ff0a - semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0 - question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0020.json)