{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0001","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcaf61913a4460c70de95a5c3fc480d8dc2a74ba5418c5e86dc2dd8c40c79207","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121001,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0002","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebb29310da838e441f36ef2edb8eabb56d8e2ef7111e3f0048837bbc972d99e2","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121002,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0003","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"916a9cd98c12815491cbf42813218dd88a571199b019baa219d75577943efa42","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121003,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0004","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc602439b8572586aa17cf9a8c942b254f7c380b49bcbd5fed8c730e9b72c2b0","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121004,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0005","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3165cc167dd5cc4421b5f2cc74c3d1967cb8060c157b25de2d609e812d51a580","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121005,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0006","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d611c4284cfb7c96f392ea598d6b802af760fae87a7f2f0cc2e51105794e41d0","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121006,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0007","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6fbbb6a3dfb6e3025ea971d38523b33d44d80cb5e71b46cf7c972965200d1cf","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121007,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0008","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fb1a2d26b77d83b7af03b03de874656503c639ca8238e1e3d3d6a1f053d5d4d","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121008,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0009","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ffae0a4b4cfff5cbce163e612d867b2ef85d670c30fefe4c7d5fbbeb2d9c531","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121009,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0010","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d9458437253025b7b53605c778f733f08f999883b49bdbda097089221882bb9","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121010,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0011","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"125eb9a7e4954bef9bd4e7f02885fb21ef4e082e22f88ee0fad0db1ba43eb5ff","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121011,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0012","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4312bbe1f9ace4efc03f0951d0a80b60fba2830b444e65f0aafdfc03cbeffcc","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121012,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0013","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d7d6c06323a581ab12f217be275bca825f5c90ada10e0ff72f939e55a2b29bf","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121013,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0014","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce591fb9ae8e86ac32f34933521f78219a144538ad25ca9de8c41d42bffb87e","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121014,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0015","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa529c0cf3b5a205dc05650fb09f503b6b6189e1c63e77d84da2a6b574f3d776","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121015,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0016","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e588b78591b9fbe5e9e4c08a523f93cf138d3732a59377822e230796a42c550c","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121016,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0017","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fcc39ea6c1daed3bee8620aee6aad392186bcd90ac08089b20d76ee1d061c48","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121017,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0018","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5518e826d62ae2beb6a702e85344cba5fe43d9eda2e153fe1cc5747fc1396aa0","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121018,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0019","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1346da6a47d01dfaefecaed37e8a7d8ee3831ce9f11a3482f8fc9f09239eb6c","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121019,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0020","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3111864bb868cecec98c2d78c2df4e5aa540c4e9bf0840023d4afe01c26bc465","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121020,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0021","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ce011f832b0efeb140efcfa812fa3352cda763a7ff2cc22dcbb8607dd690840","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121021,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0022","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8154a409ca45069f7833354859cc39ca5d9b6b2fb11e035d42838093745c47d","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121022,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(8, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0023","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eda961e11791c735d5958193388c7061f5f5cb977d74e3c5777ec81729c0881","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121023,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(10, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0024","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"139f5864e2a3773f5a5843df8a4fd5632bb63325dad6301c4b38ae3c5976b752","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121024,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(12, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0025","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90b158e1653e2f60a9a695f592287018bf6ff55aa70a49b4bf0e64e8f2719b6c","family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121025,"semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"Cauchy Requires a Prime Divisor of a Fully Certified Finite Group","stem":"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).","answer":{"kind":"expression","value":"(14, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation":"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)."},"skills":["Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0026","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6a207f5ab649ca9ebb8a73c2f51345853916d613ea966a7b2ea6f3564a366a9","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121026,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":0}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(6, [[0, 1], [1, 6], [2, 3], [3, 2], [4, 3], [5, 6]], Yes, complete_element_order_table_for_declared_finite_group)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0027","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ed3d1538bcb9efeb856ff329e9637a53697695966c0fa52348b25803eed8a5b","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121027,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":0}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0028","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3da4de7f9eef6444d6dcc9cf101a1e3638ecac0d5941484236a8d78caa9f63da","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121028,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":0}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0029","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c865695e6e8c1fa8ee313a15d33952eb9a519ead07ea32f1a4227467c97f88","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121029,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":0}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0030","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08863e3f9d4b8cd74535b10118fd436190c94e9c75a42231f7163c51a32191ba","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121030,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":0}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0031","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b32407146402857a8af3edfaf827938c1dddbe9f4ccb4862fbfeca1a36dc40a7","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121031,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":1}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(6, [[0, 6], [1, 1], [2, 6], [3, 3], [4, 2], [5, 3]], Yes, complete_element_order_table_for_declared_finite_group)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0032","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c4d7a5a52b6004cc5b0f85ca1f8ca4b4307b1932c90ac38b88d25261906e03b","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121032,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":1}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0033","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd71719bddbff2ae9b26179cad674261ddc856fcc76fd93bc8b083f253b828ab","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121033,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":1}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0034","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"553d88e3d710d9fa1a863d1309078c3ab44b2201bfe2eeb0a05c9d5859870295","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121034,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":1}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0035","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2ad5c0201a3263ed7994d79a72b2c07fc16f5c7edc4f8de49329a3e79b0bff2","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121035,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":1}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0036","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be670d8f929921efdb51350a5e07927212819c0291cfe9888e1b2b992c2d20b0","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121036,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":2}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(6, [[0, 3], [1, 6], [2, 1], [3, 6], [4, 3], [5, 2]], Yes, complete_element_order_table_for_declared_finite_group)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0037","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5d96d5fe0dfc63943138b698de7da5d1155f0ab59e4e822da04ed9a782e29de","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121037,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":2}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0038","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"817312cdc653ba866e09783b7b2e0c677ea2687776f1e7b8b805dba5dbe4aff9","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121038,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":2}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0039","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c3dcaa700f0aa358ce4ae7b4aebb95f5c2c6f3f5e690554fc999b0edea665d3","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121039,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":2}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0040","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2781ad5c745dfe145763284c0520f8381161cea088948aefdbd7861579242e1","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121040,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":2}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0041","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52bfc44bb18d80d4bf2265b5a7f3d82e39c01fb0c47b910dd8f25ad84636d779","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121041,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":3}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(6, [[0, 2], [1, 3], [2, 6], [3, 1], [4, 6], [5, 3]], Yes, complete_element_order_table_for_declared_finite_group)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0042","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffcd67c1605deb3261ce2c67b04583907be306b2cc45a82215598d699b18c5e5","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121042,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":3}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0043","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fb1aeea3f1aa8c15bc59fb437ec3d3614501dbc488c68eab7fa17cac0568118","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121043,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":3}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0044","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d15659b69f2377fbf9a05318270c028a91ebdec34ace4d47c3e7b7442edb2913","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121044,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":3}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0045","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aa0ef55ca5dac70975631d29eea4a34367071ded97913b8521115fb66fcb42f","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121045,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":3}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0046","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c22959cd56852c92f18dccfce8ff40b5dc471bf61d5f65cdcf50b04c4deb6266","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121046,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":4}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(6, [[0, 3], [1, 2], [2, 3], [3, 6], [4, 1], [5, 6]], Yes, complete_element_order_table_for_declared_finite_group)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0047","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cacfc09cc9faa6438b047820aa5b692674840a1376eae0be3c1622712081d51","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121047,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":4}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0048","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed149e909cfe3884d4cd5cba22db04d4947b658b25e12f91e7b7add061cb3d31","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121048,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":4}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0049","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5805c028c39e7768d441b346b50425ce513ce35c1697073d90d707f57ab6003","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121049,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":4}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0050","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fc94172c8fd70a9664315142b8476d175e2782e4e7118d271dc14c25d306631","family_id":"qf-mod43-cauchy-element-order-table","seed":202611121050,"semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_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]],\"identity\":4}"},"fields":{"title":"Every Element Order Is the First Positive Identity Return","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["Every Element Order Is the First Positive Identity Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0051","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af37ceb70e858fe26f38dfa9442d49d37d39c60880aa944434e3caba2219d26e","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121051,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":3,\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(6, 2, 3, [[1, 3], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0052","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef1dd9f509f96b5c79e3f3001da084fee8286d23d196e23187804d5be670ae9b","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121052,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":4,\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(8, 2, 4, [[1, 4], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0053","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ea84ed88fdde40d17c82ae7ae7504c1929965fd7380fffaa5c324b6f1aff54e","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121053,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":5,\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(10, 2, 5, [[1, 5], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0054","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6f00f8dbd914ab702377668bd270354e5b4de1427baee20e05515c5fc5631fe","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121054,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":6,\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(12, 2, 6, [[1, 6], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0055","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f3e95eab375807a4f906cd94b980797b1c4c9f6a71a6b37aba8f074dde8a53c","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121055,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"group_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]],\"identity\":0,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(14, 2, 7, [[1, 7], [2, 0]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0056","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"584545da9609dfa966a1a36129b8f701b27ddca904641947e007172c60988175","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121056,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":4,\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(6, 2, 4, [[1, 4], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0057","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ae2463a5a11eb804a237b2ecfc5d01b21e2f73c0750b033b646aea454d368d3","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121057,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":5,\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(8, 2, 5, [[1, 5], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0058","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89fb8af4bec40f7362bd65f8ca9377dab0f2ed3b3e89a32de5ec5464c9a27ce1","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121058,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":6,\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(10, 2, 6, [[1, 6], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0059","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73341c5ad0f8ca12a4c5e6699762e82ab61f9182bea57dd83fabb431d0b4aa94","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121059,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(12, 2, 7, [[1, 7], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0060","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3e0a59b1ed5aaea937e2a8538f978a0730cf75dcfabf9c06678b16f4ae89ce3","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121060,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":8,\"group_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]],\"identity\":1,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(14, 2, 8, [[1, 8], [2, 1]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0061","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d39ce9f41ed354f382d2c8a06615453fb8db860f1f81ffed8db07e3f6d7d302","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121061,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":5,\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(6, 2, 5, [[1, 5], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0062","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c5f779b3ad0c6a1b5bafb115d9121151672648faab02a741c4b9f0390955ee1","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121062,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":6,\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(8, 2, 6, [[1, 6], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0063","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbdbe5e313e94291b88d93390328031cb44fa88ceda27e2caf36e1d743ceb014","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121063,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(10, 2, 7, [[1, 7], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0064","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2b4e6470bca0eadf5869d357da6b4dbe58257288909cbd7a10b32e953cf8089","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121064,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":8,\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(12, 2, 8, [[1, 8], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0065","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9848ca7efc322ae9c1e6c3c4a1d118dcbc427849bea93ba9320935c1b9a723e8","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121065,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":9,\"group_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]],\"identity\":2,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(14, 2, 9, [[1, 9], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0066","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67b62aef79aefc92210c8f08c792a880cb38e675392f31594b31d6aa53d5022c","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121066,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":0,\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(6, 2, 0, [[1, 0], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0067","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05daa75bc5ea729aec525a6702fe198859603ffe5e76880df8792d51d6c9f183","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121067,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(8, 2, 7, [[1, 7], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0068","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b3faafb33d3e5c99e70139d669e733d12174e39d34a52c1c5b7f43db466007d","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121068,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":8,\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(10, 2, 8, [[1, 8], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0069","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24c8cf80d4d0612e0c9ae689a2b1d1af1f90ffdf0f0ee2286b1cc84a082f9dc2","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121069,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":9,\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(12, 2, 9, [[1, 9], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0070","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd716f149214f4f01522b6f4f60e05d2316b1feaee5b68cafa87e72762c296f1","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121070,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":10,\"group_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]],\"identity\":3,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(14, 2, 10, [[1, 10], [2, 3]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0071","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f82e998f8636d8b7de191bd52a852e83e41eb5422f4570e5c90a44a005ae791b","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121071,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":1,\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(6, 2, 1, [[1, 1], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0072","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a53712ea4925b7b12bcdc3c7d3f9b5406f01a9e11a7a82a2b2ea26f4ecaec39","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121072,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":0,\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(8, 2, 0, [[1, 0], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0073","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"197d64028e1929e28ac1d331fc18d4615aaef8090fc23c2be7978f37980d0b9d","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121073,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":9,\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(10, 2, 9, [[1, 9], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0074","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3b6843f258d4eb143522bc76cd105f4482d2f625c3d0c22eaf10c01d2b42421","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121074,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":10,\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(12, 2, 10, [[1, 10], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0075","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"605416756fcfcd5beec9ee1976af93c6ecbcb0644bf7c49306d9f8fb247aed01","family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121075,"semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":11,\"group_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]],\"identity\":4,\"prime\":2}"},"fields":{"title":"An Order-p Witness Must Exclude Every Smaller Positive Return","stem":"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).","answer":{"kind":"expression","value":"(14, 2, 11, [[1, 11], [2, 4]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation":"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)."},"skills":["An Order-p Witness Must Exclude Every Smaller Positive Return"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0076","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c26e205403faf9dca849c40f6614e7524a2d8e047432ea7e9bfda1180e620ff9","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121076,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,0,1],[3,2,1,0]],\"identity\":0}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(4, 4, [[0, 1], [1, 2], [2, 2], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0077","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e525e6dc082fba7ea79af59ca949f4bbc10fd23253b07c8e5b46de9e7be0f357","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121077,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[1,0,3,2],[0,1,2,3],[3,2,1,0],[2,3,0,1]],\"identity\":1}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(4, 4, [[0, 2], [1, 1], [2, 2], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0078","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c147c6eede49ea0b172b2e2cec17a937ab91634dcaeff314347a28c2cd3c6df5","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121078,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[2,3,0,1],[3,2,1,0],[0,1,2,3],[1,0,3,2]],\"identity\":2}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(4, 4, [[0, 2], [1, 2], [2, 1], [3, 2]], Yes, No, composite_divisor_need_not_be_an_element_order)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0079","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0694821a20c9b48051d5500aa4fd6797fbee430798869ecbbd31d72375d44523","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121079,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[3,2,1,0],[2,3,0,1],[1,0,3,2],[0,1,2,3]],\"identity\":3}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(4, 4, [[0, 2], [1, 2], [2, 2], [3, 1]], Yes, No, composite_divisor_need_not_be_an_element_order)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0080","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84dec78630f6c006dbd5778831631d8905075da8db5dedb5168932677344bd54","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121080,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":0}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0081","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d34e242308fc9b6a7567b5d5b08958605ac9795eb54d6315b9d6614a96e63f8c","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121081,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":1}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0082","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"340dcbae846989a61b42d7b11e648c3b8145a8c9b3087ac414a4e988fce648b6","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121082,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":2}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0083","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b25b500019846d3ae3959a9d9f935c3b8cfcfb848ad1ff59427db2dca030fb46","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121083,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":3}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0084","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b17ccfe7ff8d5ff3218fe47e33846262857e657272442b0fec8358f7a53d3c9","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121084,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":4}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0085","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07a79b7a527c5a08fb41457cacf790320c96955139caa0dea9073e2645f81d66","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121085,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":5}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0086","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9923a1d774af360810f7193ceab6db1c8bbc1da6cc8d04c906ef909d8c09d077","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121086,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":6}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0087","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3238a1d89723b3ed4c6a8f642659a17f68752704aa7e26365dada7005d1699d8","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121087,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_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]],\"identity\":7}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0088","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2055837eb8dcde6efc8fb604170e92dad9f11e11fd24a817d2f9123101e5048e","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121088,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":0}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0089","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1774770336fe71e763aef3b67bce318fa22881355279f8e9ba78e2e9a474265b","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121089,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":1}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0090","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9a56760ef1dda17e08d088f1653e2f2167dcfadb23ff540af8e2b8a97a845c3","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121090,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":2}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0091","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e33a73a9daee7c689a2af2a10b962c1f2901235ae0bf6ebf2c1e60d0f2ce5b25","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121091,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":3}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0092","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91a25ada58a20e908de421a34069110b4f8f58657c7d625e87be950ec20ed6a1","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121092,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":4}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0093","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60c62b50033e9b352c5fd96915139e9b9d4d5d02cebc10c3173f48dd51020fac","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121093,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":5}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0094","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b33b8cfce6d66a92d855dfffc741b9dbcaf2b03f80fd7f92ebce32f74a7aca","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121094,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":6}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0095","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06554e5b949056a8273d9386f2ac16641995a4b3ed7cd70fcbf80e72546f6f66","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121095,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":7}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0096","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"582eeab45ca2367f3b9ccfe27a30df8ba3b84f762fa4c82003d150fc7d2df783","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121096,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":8}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0097","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"631170233b2becddd229714e10333433d9424f49c6c624bc9242daff9d8a223d","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121097,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":9}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0098","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"231fe7f889ae40dc9acd553a38986c18b54463bda0aa5371190638176b1ac196","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121098,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":10}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0099","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38e5b49969ef5ee37457db8b9dfddb349eb77f938291a79ef7cdcfb3e3e454de","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121099,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":11}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0100","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb4b9a824adaeb9514bdf16102fd9316dab2eeef6b0147d98554d67903d16f13","family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121100,"semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_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]],\"identity\":12}"},"fields":{"title":"A Composite Divisor Need Not Be an Element Order","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Composite Divisor Need Not Be an Element Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}