Prime Divisors, Exact Element Orders and Composite-Order Nonconverses
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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-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite 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]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
2. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
3. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
4. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
5. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
6. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[5, 0, 1, 2, 3, 4], [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]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
7. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
8. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
9. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
10. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
11. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
12. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
13. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
14. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
15. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
16. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
17. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
18. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
19. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
20. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
21. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
22. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
23. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
24. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
25. Cauchy Requires a Prime Divisor of a Fully Certified Finite Group
Use complete finite Cayley table=[[10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).
View answer and explanation
(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)
Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-prime-divisor-qualification · candidate
26. Every Element Order Is the First Positive Identity Return
Use complete finite 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]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(6, [[0, 1], [1, 6], [2, 3], [3, 2], [4, 3], [5, 6]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (6, [[0, 1], [1, 6], [2, 3], [3, 2], [4, 3], [5, 6]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
27. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(8, [[0, 1], [1, 8], [2, 4], [3, 8], [4, 2], [5, 8], [6, 4], [7, 8]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (8, [[0, 1], [1, 8], [2, 4], [3, 8], [4, 2], [5, 8], [6, 4], [7, 8]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
28. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(10, [[0, 1], [1, 10], [2, 5], [3, 10], [4, 5], [5, 2], [6, 5], [7, 10], [8, 5], [9, 10]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (10, [[0, 1], [1, 10], [2, 5], [3, 10], [4, 5], [5, 2], [6, 5], [7, 10], [8, 5], [9, 10]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
29. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(12, [[0, 1], [1, 12], [2, 6], [3, 4], [4, 3], [5, 12], [6, 2], [7, 12], [8, 3], [9, 4], [10, 6], [11, 12]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (12, [[0, 1], [1, 12], [2, 6], [3, 4], [4, 3], [5, 12], [6, 2], [7, 12], [8, 3], [9, 4], [10, 6], [11, 12]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
30. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(14, [[0, 1], [1, 14], [2, 7], [3, 14], [4, 7], [5, 14], [6, 7], [7, 2], [8, 7], [9, 14], [10, 7], [11, 14], [12, 7], [13, 14]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (14, [[0, 1], [1, 14], [2, 7], [3, 14], [4, 7], [5, 14], [6, 7], [7, 2], [8, 7], [9, 14], [10, 7], [11, 14], [12, 7], [13, 14]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
31. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[5, 0, 1, 2, 3, 4], [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]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(6, [[0, 6], [1, 1], [2, 6], [3, 3], [4, 2], [5, 3]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (6, [[0, 6], [1, 1], [2, 6], [3, 3], [4, 2], [5, 3]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
32. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(8, [[0, 8], [1, 1], [2, 8], [3, 4], [4, 8], [5, 2], [6, 8], [7, 4]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (8, [[0, 8], [1, 1], [2, 8], [3, 4], [4, 8], [5, 2], [6, 8], [7, 4]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
33. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(10, [[0, 10], [1, 1], [2, 10], [3, 5], [4, 10], [5, 5], [6, 2], [7, 5], [8, 10], [9, 5]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (10, [[0, 10], [1, 1], [2, 10], [3, 5], [4, 10], [5, 5], [6, 2], [7, 5], [8, 10], [9, 5]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
34. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(12, [[0, 12], [1, 1], [2, 12], [3, 6], [4, 4], [5, 3], [6, 12], [7, 2], [8, 12], [9, 3], [10, 4], [11, 6]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (12, [[0, 12], [1, 1], [2, 12], [3, 6], [4, 4], [5, 3], [6, 12], [7, 2], [8, 12], [9, 3], [10, 4], [11, 6]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
35. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(14, [[0, 14], [1, 1], [2, 14], [3, 7], [4, 14], [5, 7], [6, 14], [7, 7], [8, 2], [9, 7], [10, 14], [11, 7], [12, 14], [13, 7]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (14, [[0, 14], [1, 1], [2, 14], [3, 7], [4, 14], [5, 7], [6, 14], [7, 7], [8, 2], [9, 7], [10, 14], [11, 7], [12, 14], [13, 7]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
36. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(6, [[0, 3], [1, 6], [2, 1], [3, 6], [4, 3], [5, 2]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (6, [[0, 3], [1, 6], [2, 1], [3, 6], [4, 3], [5, 2]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
37. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(8, [[0, 4], [1, 8], [2, 1], [3, 8], [4, 4], [5, 8], [6, 2], [7, 8]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (8, [[0, 4], [1, 8], [2, 1], [3, 8], [4, 4], [5, 8], [6, 2], [7, 8]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
38. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(10, [[0, 5], [1, 10], [2, 1], [3, 10], [4, 5], [5, 10], [6, 5], [7, 2], [8, 5], [9, 10]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (10, [[0, 5], [1, 10], [2, 1], [3, 10], [4, 5], [5, 10], [6, 5], [7, 2], [8, 5], [9, 10]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
39. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(12, [[0, 6], [1, 12], [2, 1], [3, 12], [4, 6], [5, 4], [6, 3], [7, 12], [8, 2], [9, 12], [10, 3], [11, 4]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (12, [[0, 6], [1, 12], [2, 1], [3, 12], [4, 6], [5, 4], [6, 3], [7, 12], [8, 2], [9, 12], [10, 3], [11, 4]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
40. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(14, [[0, 7], [1, 14], [2, 1], [3, 14], [4, 7], [5, 14], [6, 7], [7, 14], [8, 7], [9, 2], [10, 7], [11, 14], [12, 7], [13, 14]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (14, [[0, 7], [1, 14], [2, 1], [3, 14], [4, 7], [5, 14], [6, 7], [7, 14], [8, 7], [9, 2], [10, 7], [11, 14], [12, 7], [13, 14]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
41. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(6, [[0, 2], [1, 3], [2, 6], [3, 1], [4, 6], [5, 3]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (6, [[0, 2], [1, 3], [2, 6], [3, 1], [4, 6], [5, 3]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
42. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(8, [[0, 8], [1, 4], [2, 8], [3, 1], [4, 8], [5, 4], [6, 8], [7, 2]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (8, [[0, 8], [1, 4], [2, 8], [3, 1], [4, 8], [5, 4], [6, 8], [7, 2]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
43. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(10, [[0, 10], [1, 5], [2, 10], [3, 1], [4, 10], [5, 5], [6, 10], [7, 5], [8, 2], [9, 5]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (10, [[0, 10], [1, 5], [2, 10], [3, 1], [4, 10], [5, 5], [6, 10], [7, 5], [8, 2], [9, 5]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
44. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(12, [[0, 4], [1, 6], [2, 12], [3, 1], [4, 12], [5, 6], [6, 4], [7, 3], [8, 12], [9, 2], [10, 12], [11, 3]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (12, [[0, 4], [1, 6], [2, 12], [3, 1], [4, 12], [5, 6], [6, 4], [7, 3], [8, 12], [9, 2], [10, 12], [11, 3]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
45. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(14, [[0, 14], [1, 7], [2, 14], [3, 1], [4, 14], [5, 7], [6, 14], [7, 7], [8, 14], [9, 7], [10, 2], [11, 7], [12, 14], [13, 7]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (14, [[0, 14], [1, 7], [2, 14], [3, 1], [4, 14], [5, 7], [6, 14], [7, 7], [8, 14], [9, 7], [10, 2], [11, 7], [12, 14], [13, 7]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
46. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(6, [[0, 3], [1, 2], [2, 3], [3, 6], [4, 1], [5, 6]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (6, [[0, 3], [1, 2], [2, 3], [3, 6], [4, 1], [5, 6]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
47. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(8, [[0, 2], [1, 8], [2, 4], [3, 8], [4, 1], [5, 8], [6, 4], [7, 8]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (8, [[0, 2], [1, 8], [2, 4], [3, 8], [4, 1], [5, 8], [6, 4], [7, 8]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
48. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(10, [[0, 5], [1, 10], [2, 5], [3, 10], [4, 1], [5, 10], [6, 5], [7, 10], [8, 5], [9, 2]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (10, [[0, 5], [1, 10], [2, 5], [3, 10], [4, 1], [5, 10], [6, 5], [7, 10], [8, 5], [9, 2]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
49. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(12, [[0, 3], [1, 4], [2, 6], [3, 12], [4, 1], [5, 12], [6, 6], [7, 4], [8, 3], [9, 12], [10, 2], [11, 12]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (12, [[0, 3], [1, 4], [2, 6], [3, 12], [4, 1], [5, 12], [6, 6], [7, 4], [8, 3], [9, 12], [10, 2], [11, 12]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
50. Every Element Order Is the First Positive Identity Return
Use complete finite Cayley table=[[10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Give (group_order, element_order_rows, all_orders_divide_group_order, certificate).
View answer and explanation
(14, [[0, 7], [1, 14], [2, 7], [3, 14], [4, 1], [5, 14], [6, 7], [7, 14], [8, 7], [9, 14], [10, 7], [11, 2], [12, 7], [13, 14]], Yes, complete_element_order_table_for_declared_finite_group)
Enumerate every element and its least positive identity-return exponent, checking through the full finite group order. Report whether every computed order divides |G|. Do not count exponent0 or infer exact order from only g^p=e. The exact result, retaining the requested order, is (14, [[0, 7], [1, 14], [2, 7], [3, 14], [4, 1], [5, 14], [6, 7], [7, 14], [8, 7], [9, 14], [10, 7], [11, 2], [12, 7], [13, 14]], Yes, complete_element_order_table_for_declared_finite_group).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-element-order-table · candidate
51. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite 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]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=3. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(6, 2, 3, [[1, 3], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (6, 2, 3, [[1, 3], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
52. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=4. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(8, 2, 4, [[1, 4], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (8, 2, 4, [[1, 4], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
53. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=5. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(10, 2, 5, [[1, 5], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (10, 2, 5, [[1, 5], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
54. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=6. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(12, 2, 6, [[1, 6], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (12, 2, 6, [[1, 6], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
55. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=7. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(14, 2, 7, [[1, 7], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (14, 2, 7, [[1, 7], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
56. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[5, 0, 1, 2, 3, 4], [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]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=4. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(6, 2, 4, [[1, 4], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (6, 2, 4, [[1, 4], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
57. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=5. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(8, 2, 5, [[1, 5], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (8, 2, 5, [[1, 5], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
58. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=6. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(10, 2, 6, [[1, 6], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (10, 2, 6, [[1, 6], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
59. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=7. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(12, 2, 7, [[1, 7], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (12, 2, 7, [[1, 7], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
60. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=8. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(14, 2, 8, [[1, 8], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (14, 2, 8, [[1, 8], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
61. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=5. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(6, 2, 5, [[1, 5], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (6, 2, 5, [[1, 5], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
62. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=6. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(8, 2, 6, [[1, 6], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (8, 2, 6, [[1, 6], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
63. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=7. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(10, 2, 7, [[1, 7], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (10, 2, 7, [[1, 7], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
64. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=8. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(12, 2, 8, [[1, 8], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (12, 2, 8, [[1, 8], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
65. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=9. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(14, 2, 9, [[1, 9], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (14, 2, 9, [[1, 9], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
66. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=0. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(6, 2, 0, [[1, 0], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (6, 2, 0, [[1, 0], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
67. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=7. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(8, 2, 7, [[1, 7], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (8, 2, 7, [[1, 7], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
68. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=8. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(10, 2, 8, [[1, 8], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (10, 2, 8, [[1, 8], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
69. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=9. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(12, 2, 9, [[1, 9], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (12, 2, 9, [[1, 9], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
70. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=10. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(14, 2, 10, [[1, 10], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (14, 2, 10, [[1, 10], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
71. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=1. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(6, 2, 1, [[1, 1], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (6, 2, 1, [[1, 1], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
72. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=0. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(8, 2, 0, [[1, 0], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (8, 2, 0, [[1, 0], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
73. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=9. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(10, 2, 9, [[1, 9], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (10, 2, 9, [[1, 9], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
74. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=10. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(12, 2, 10, [[1, 10], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (12, 2, 10, [[1, 10], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
75. An Order-p Witness Must Exclude Every Smaller Positive Return
Use complete finite Cayley table=[[10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=11. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).
Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness.
View answer and explanation
(14, 2, 11, [[1, 11], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)
Require prime divisibility and exact element order p. List powers1 through p, verify the p-th is identity and every smaller positive one is not. The identity is not an order-p witness even though its p-th power is identity. The exact result, retaining the requested order, is (14, 2, 11, [[1, 11], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-order-p-witness · candidate
76. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 0, 1], [3, 2, 1, 0]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=4. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(4, 4, [[0, 1], [1, 2], [2, 2], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (4, 4, [[0, 1], [1, 2], [2, 2], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
77. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 1, 0], [2, 3, 0, 1]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=4. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(4, 4, [[0, 2], [1, 1], [2, 2], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (4, 4, [[0, 2], [1, 1], [2, 2], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
78. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[2, 3, 0, 1], [3, 2, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=4. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(4, 4, [[0, 2], [1, 2], [2, 1], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (4, 4, [[0, 2], [1, 2], [2, 1], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
79. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[3, 2, 1, 0], [2, 3, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=4. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(4, 4, [[0, 2], [1, 2], [2, 2], [3, 1]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (4, 4, [[0, 2], [1, 2], [2, 2], [3, 1]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
80. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 0, 3, 2, 5, 4, 7, 6], [2, 3, 0, 1, 6, 7, 4, 5], [3, 2, 1, 0, 7, 6, 5, 4], [4, 5, 6, 7, 0, 1, 2, 3], [5, 4, 7, 6, 1, 0, 3, 2], [6, 7, 4, 5, 2, 3, 0, 1], [7, 6, 5, 4, 3, 2, 1, 0]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
81. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7], [3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5], [5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3], [7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
82. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[2, 3, 0, 1, 6, 7, 4, 5], [3, 2, 1, 0, 7, 6, 5, 4], [0, 1, 2, 3, 4, 5, 6, 7], [1, 0, 3, 2, 5, 4, 7, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 6, 5, 4, 3, 2, 1, 0], [4, 5, 6, 7, 0, 1, 2, 3], [5, 4, 7, 6, 1, 0, 3, 2]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
83. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5], [1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7], [7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1], [5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
84. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[4, 5, 6, 7, 0, 1, 2, 3], [5, 4, 7, 6, 1, 0, 3, 2], [6, 7, 4, 5, 2, 3, 0, 1], [7, 6, 5, 4, 3, 2, 1, 0], [0, 1, 2, 3, 4, 5, 6, 7], [1, 0, 3, 2, 5, 4, 7, 6], [2, 3, 0, 1, 6, 7, 4, 5], [3, 2, 1, 0, 7, 6, 5, 4]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
85. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3], [7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1], [1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7], [3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5]] with identity=5, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
86. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[6, 7, 4, 5, 2, 3, 0, 1], [7, 6, 5, 4, 3, 2, 1, 0], [4, 5, 6, 7, 0, 1, 2, 3], [5, 4, 7, 6, 1, 0, 3, 2], [2, 3, 0, 1, 6, 7, 4, 5], [3, 2, 1, 0, 7, 6, 5, 4], [0, 1, 2, 3, 4, 5, 6, 7], [1, 0, 3, 2, 5, 4, 7, 6]] with identity=6, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
87. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[7, 6, 5, 4, 3, 2, 1, 0], [6, 7, 4, 5, 2, 3, 0, 1], [5, 4, 7, 6, 1, 0, 3, 2], [4, 5, 6, 7, 0, 1, 2, 3], [3, 2, 1, 0, 7, 6, 5, 4], [2, 3, 0, 1, 6, 7, 4, 5], [1, 0, 3, 2, 5, 4, 7, 6], [0, 1, 2, 3, 4, 5, 6, 7]] with identity=7, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=8. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
88. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]] with identity=0, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
89. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1]] with identity=1, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
90. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
91. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
92. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4]] with identity=4, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
93. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5]] with identity=5, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
94. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6]] with identity=6, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
95. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7]] with identity=7, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
96. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8]] with identity=8, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 1], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 1], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
97. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9]] with identity=9, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 1], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 1], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
98. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10]] with identity=10, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 1], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 1], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
99. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11]] with identity=11, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 1], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 1], [12, 2], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
100. A Composite Divisor Need Not Be an Element Order
Use complete finite Cayley table=[[12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3], [13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2], [14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6], [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10], [6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9], [7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14], [2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13], [3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12]] with identity=12, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Requested composite divisor=16. Give (group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary).
View answer and explanation
(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 1], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)
Require actual divisibility and exhaust the entire element-order table with no element of that requested order. The final No is absence in this complete model, not a claim that composite element orders never occur in other groups. The exact result, retaining the requested order, is (16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 1], [13, 2], [14, 2], [15, 2]], Yes, No, composite_divisor_need_not_be_an_element_order).
Reviewed step-by-step derivation
Cauchy theorem for finite groups · qf-mod43-cauchy-composite-divisor-boundary · candidate
Canonical question JSONL · English JSONL · Validation report