Qualified Finite Groups, Commutativity, Orders and Cyclic Subgroups
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261026-r01-finite-abelian-group-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
2. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
3. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
4. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
5. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
6. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
7. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
8. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
9. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
10. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
11. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
12. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
13. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
14. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
15. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
16. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
17. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
18. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
19. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
20. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
21. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
22. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
23. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
24. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
25. Audit Every Pair of a Qualified Finite Group
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).
View answer and explanation
(6, 0, [], Yes)
Compare both operation orders for every distinct unordered label pair. The diagonal commutes automatically, but does not establish the off-diagonal obligations. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, [], Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-commutativity-audit · candidate
26. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(1, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
27. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(0, 0, 1, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
28. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(5, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
29. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(4, 0, 2, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 2, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
30. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(3, 0, 3, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 3, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
31. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(2, 0, 3, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 3, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
32. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(1, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
33. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(0, 0, 1, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
34. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(5, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
35. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(4, 0, 2, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 2, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
36. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(3, 0, 3, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 3, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
37. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(2, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
38. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(1, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
39. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(0, 0, 1, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
40. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(5, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
41. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(4, 0, 3, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 3, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
42. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(3, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
43. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(2, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
44. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(1, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
45. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(0, 0, 1, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 1, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
46. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(5, 0, 3, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 3, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
47. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(4, 0, 3, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 3, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
48. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(3, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
49. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(2, 0, 6, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 6, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
50. Element Order Starts from the Actual Identity
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,least positive order,returns to identity?).
View answer and explanation
(1, 0, 3, Yes)
Iterate the given operation from its qualified identity until the first positive return; do not substitute an assumed zero label. This calculation does not itself assert the group is abelian. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 3, Yes).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-element-order · candidate
51. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=3. Give (element,multiplier,identity,result).
View answer and explanation
(1, 3, 0, 3)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3, 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
52. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=4. Give (element,multiplier,identity,result).
View answer and explanation
(0, 4, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 4, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
53. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=5. Give (element,multiplier,identity,result).
View answer and explanation
(5, 5, 0, 1)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 5, 0, 1).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
54. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=6. Give (element,multiplier,identity,result).
View answer and explanation
(4, 6, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 6, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
55. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=7. Give (element,multiplier,identity,result).
View answer and explanation
(3, 7, 0, 3)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 7, 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
56. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=8. Give (element,multiplier,identity,result).
View answer and explanation
(2, 8, 0, 4)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 8, 0, 4).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
57. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=9. Give (element,multiplier,identity,result).
View answer and explanation
(1, 9, 0, 2)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 9, 0, 2).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
58. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=10. Give (element,multiplier,identity,result).
View answer and explanation
(0, 10, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 10, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
59. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=11. Give (element,multiplier,identity,result).
View answer and explanation
(5, 11, 0, 1)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 11, 0, 1).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
60. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=12. Give (element,multiplier,identity,result).
View answer and explanation
(4, 12, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 12, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
61. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=13. Give (element,multiplier,identity,result).
View answer and explanation
(3, 13, 0, 3)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 13, 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
62. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=14. Give (element,multiplier,identity,result).
View answer and explanation
(2, 14, 0, 4)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 14, 0, 4).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
63. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=15. Give (element,multiplier,identity,result).
View answer and explanation
(1, 15, 0, 2)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 15, 0, 2).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
64. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=16. Give (element,multiplier,identity,result).
View answer and explanation
(0, 16, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 16, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
65. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=17. Give (element,multiplier,identity,result).
View answer and explanation
(5, 17, 0, 1)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 17, 0, 1).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
66. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=18. Give (element,multiplier,identity,result).
View answer and explanation
(4, 18, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 18, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
67. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=19. Give (element,multiplier,identity,result).
View answer and explanation
(3, 19, 0, 3)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 19, 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
68. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=20. Give (element,multiplier,identity,result).
View answer and explanation
(2, 20, 0, 3)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 20, 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
69. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=21. Give (element,multiplier,identity,result).
View answer and explanation
(1, 21, 0, 2)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 21, 0, 2).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
70. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0, nonnegative multiplier=22. Give (element,multiplier,identity,result).
View answer and explanation
(0, 22, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 22, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
71. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5, nonnegative multiplier=23. Give (element,multiplier,identity,result).
View answer and explanation
(5, 23, 0, 2)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 23, 0, 2).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
72. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4, nonnegative multiplier=24. Give (element,multiplier,identity,result).
View answer and explanation
(4, 24, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 24, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
73. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3, nonnegative multiplier=25. Give (element,multiplier,identity,result).
View answer and explanation
(3, 25, 0, 3)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 25, 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
74. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2, nonnegative multiplier=26. Give (element,multiplier,identity,result).
View answer and explanation
(2, 26, 0, 3)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 26, 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
75. Repeated Group Operation with a Zero-Multiple Boundary
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1, nonnegative multiplier=27. Give (element,multiplier,identity,result).
View answer and explanation
(1, 27, 0, 0)
Apply the operation the declared number of times from the identity. Zero repetitions give the actual identity, not necessarily label zero. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 27, 0, 0).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-integer-multiple · candidate
76. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(1, 0, [0, 1, 2, 3, 4, 5], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 2, 3, 4, 5], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
77. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(0, 0, [0], 0, 1)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
78. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(5, 0, [0, 5, 3, 4, 2, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 3, 4, 2, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
79. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(4, 0, [0, 4], 0, 2)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4], 0, 2).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
80. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(3, 0, [0, 3, 2], 0, 3)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2], 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
81. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(2, 0, [0, 2, 4], 0, 3)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 4], 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
82. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(1, 0, [0, 1, 3, 2, 4, 5], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 3, 2, 4, 5], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
83. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(0, 0, [0], 0, 1)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
84. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(5, 0, [0, 5, 2, 4, 3, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 2, 4, 3, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
85. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(4, 0, [0, 4], 0, 2)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4], 0, 2).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
86. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(3, 0, [0, 3, 2], 0, 3)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2], 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
87. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(2, 0, [0, 2, 4, 5, 3, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 4, 5, 3, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
88. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(1, 0, [0, 1, 4, 2, 3, 5], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 4, 2, 3, 5], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
89. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(0, 0, [0], 0, 1)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
90. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(5, 0, [0, 5, 2, 3, 4, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 2, 3, 4, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
91. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(4, 0, [0, 4, 5], 0, 3)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4, 5], 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
92. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(3, 0, [0, 3, 2, 5, 4, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2, 5, 4, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
93. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(2, 0, [0, 2, 3, 5, 4, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 3, 5, 4, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
94. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(1, 0, [0, 1, 5, 2, 3, 4], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 5, 2, 3, 4], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
95. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=0. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(0, 0, [0], 0, 1)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0], 0, 1).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
96. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(5, 0, [0, 5, 2], 0, 3)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, [0, 5, 2], 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
97. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=4. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(4, 0, [0, 4, 5], 0, 3)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, [0, 4, 5], 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
98. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(3, 0, [0, 3, 2, 4, 5, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, [0, 3, 2, 4, 5, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
99. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=2. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(2, 0, [0, 2, 3, 4, 5, 1], 0, 6)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 2, 3, 4, 5, 1], 0, 6).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
100. Preserve the Generation Order of One Cyclic Subgroup
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=1. Give (element,identity,generated elements in iteration order,next repeated value,order).
View answer and explanation
(1, 0, [0, 1, 4], 0, 3)
Begin at the identity, retain each new power in encounter order and stop at the first repeat. Do not sort this cycle as a label set or infer that it is the entire group. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [0, 1, 4], 0, 3).
Reviewed step-by-step derivation
Finite abelian group · qf-mod26-abelian-cyclic-subgroup · candidate
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