Qualified Quotient-to-Image Isomorphisms and Codomain Boundaries
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(4, 2, [0, 1, 0, 1], Yes, null, [0, 2], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (4, 2, [0, 1, 0, 1], Yes, null, [0, 2], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
2. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(4, 4, [0, 1, 2, 3], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (4, 4, [0, 1, 2, 3], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
3. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(4, 4, [0, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (4, 4, [0, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
4. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(5, 5, [0, 1, 2, 3, 4], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 1, 2, 3, 4], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
5. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 2, 4, 1, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(5, 5, [0, 2, 4, 1, 3], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 2, 4, 1, 3], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
6. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(5, 5, [0, 3, 1, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 3, 1, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
7. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(5, 5, [0, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
8. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(6, 2, [0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 2, [0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
9. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 1, 2, 0, 1, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(6, 3, [0, 1, 2, 0, 1, 2], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 3, [0, 1, 2, 0, 1, 2], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
10. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 2, 1, 0, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(6, 3, [0, 2, 1, 0, 2, 1], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 3, [0, 2, 1, 0, 2, 1], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
11. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 1, 2, 3, 4, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(6, 6, [0, 1, 2, 3, 4, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 6, [0, 1, 2, 3, 4, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
12. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(6, 6, [0, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 6, [0, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
13. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(7, 7, [0, 1, 2, 3, 4, 5, 6], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 1, 2, 3, 4, 5, 6], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
14. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 1, 3, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(7, 7, [0, 2, 4, 6, 1, 3, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 2, 4, 6, 1, 3, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
15. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 3, 6, 2, 5, 1, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(7, 7, [0, 3, 6, 2, 5, 1, 4], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 3, 6, 2, 5, 1, 4], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
16. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 4, 1, 5, 2, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(7, 7, [0, 4, 1, 5, 2, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 4, 1, 5, 2, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
17. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 5, 3, 1, 6, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(7, 7, [0, 5, 3, 1, 6, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 5, 3, 1, 6, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
18. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(7, 7, [0, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
19. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(8, 2, [0, 1, 0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4, 6], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 2, [0, 1, 0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4, 6], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
20. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(8, 4, [0, 1, 2, 3, 0, 1, 2, 3], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 4, [0, 1, 2, 3, 0, 1, 2, 3], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
21. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1, 0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(8, 4, [0, 3, 2, 1, 0, 3, 2, 1], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 4, [0, 3, 2, 1, 0, 3, 2, 1], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
22. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(8, 8, [0, 1, 2, 3, 4, 5, 6, 7], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 1, 2, 3, 4, 5, 6, 7], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
23. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 3, 6, 1, 4, 7, 2, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(8, 8, [0, 3, 6, 1, 4, 7, 2, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 3, 6, 1, 4, 7, 2, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
24. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 5, 2, 7, 4, 1, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(8, 8, [0, 5, 2, 7, 4, 1, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 5, 2, 7, 4, 1, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
25. Homomorphism, Kernel and Image Qualification Precede First Isomorphism
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).
View answer and explanation
(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)
Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 7, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-complete-qualification · candidate
26. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0, 2], [[0, 2], [1, 3]], [[0, [0, 2], [0], Yes], [1, [1, 3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 2], [[0, 2], [1, 3]], [[0, [0, 2], [0], Yes], [1, [1, 3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
27. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
28. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [2], Yes], [3, [3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [2], Yes], [3, [3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
29. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
30. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 2, 4, 1, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [1], Yes], [4, [4], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [1], Yes], [4, [4], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
31. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [1], Yes], [3, [3], [4], Yes], [4, [4], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [1], Yes], [3, [3], [4], Yes], [4, [4], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
32. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [3], Yes], [3, [3], [2], Yes], [4, [4], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [3], Yes], [3, [3], [2], Yes], [4, [4], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
33. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0, 2, 4], [[0, 2, 4], [1, 3, 5]], [[0, [0, 2, 4], [0], Yes], [1, [1, 3, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 2, 4], [[0, 2, 4], [1, 3, 5]], [[0, [0, 2, 4], [0], Yes], [1, [1, 3, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
34. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 1, 2, 0, 1, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [1], Yes], [2, [2, 5], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [1], Yes], [2, [2, 5], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
35. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 2, 1, 0, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [2], Yes], [2, [2, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [2], Yes], [2, [2, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
36. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 1, 2, 3, 4, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
37. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [4], Yes], [3, [3], [3], Yes], [4, [4], [2], Yes], [5, [5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [4], Yes], [3, [3], [3], Yes], [4, [4], [2], Yes], [5, [5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
38. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
39. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 1, 3, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [6], Yes], [4, [4], [1], Yes], [5, [5], [3], Yes], [6, [6], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [6], Yes], [4, [4], [1], Yes], [5, [5], [3], Yes], [6, [6], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
40. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 3, 6, 2, 5, 1, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [2], Yes], [4, [4], [5], Yes], [5, [5], [1], Yes], [6, [6], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [2], Yes], [4, [4], [5], Yes], [5, [5], [1], Yes], [6, [6], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
41. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 4, 1, 5, 2, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [1], Yes], [3, [3], [5], Yes], [4, [4], [2], Yes], [5, [5], [6], Yes], [6, [6], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [1], Yes], [3, [3], [5], Yes], [4, [4], [2], Yes], [5, [5], [6], Yes], [6, [6], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
42. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 5, 3, 1, 6, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [3], Yes], [3, [3], [1], Yes], [4, [4], [6], Yes], [5, [5], [4], Yes], [6, [6], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [3], Yes], [3, [3], [1], Yes], [4, [4], [6], Yes], [5, [5], [4], Yes], [6, [6], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
43. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [6], Yes], [2, [2], [5], Yes], [3, [3], [4], Yes], [4, [4], [3], Yes], [5, [5], [2], Yes], [6, [6], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [6], Yes], [2, [2], [5], Yes], [3, [3], [4], Yes], [4, [4], [3], Yes], [5, [5], [2], Yes], [6, [6], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
44. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], [[0, [0, 2, 4, 6], [0], Yes], [1, [1, 3, 5, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], [[0, [0, 2, 4, 6], [0], Yes], [1, [1, 3, 5, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
45. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [1], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [1], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
46. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1, 0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [3], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [3], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
47. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes], [7, [7], [7], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes], [7, [7], [7], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
48. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 3, 6, 1, 4, 7, 2, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [1], Yes], [4, [4], [4], Yes], [5, [5], [7], Yes], [6, [6], [2], Yes], [7, [7], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [1], Yes], [4, [4], [4], Yes], [5, [5], [7], Yes], [6, [6], [2], Yes], [7, [7], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
49. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 5, 2, 7, 4, 1, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [2], Yes], [3, [3], [7], Yes], [4, [4], [4], Yes], [5, [5], [1], Yes], [6, [6], [6], Yes], [7, [7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [2], Yes], [3, [3], [7], Yes], [4, [4], [4], Yes], [5, [5], [1], Yes], [6, [6], [6], Yes], [7, [7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
50. The Induced Map Must Agree on Every Kernel-Coset Representative
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).
View answer and explanation
([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [7], Yes], [2, [2], [6], Yes], [3, [3], [5], Yes], [4, [4], [4], Yes], [5, [5], [3], Yes], [6, [6], [2], Yes], [7, [7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)
Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [7], Yes], [2, [2], [6], Yes], [3, [3], [5], Yes], [4, [4], [4], Yes], [5, [5], [3], Yes], [6, [6], [2], Yes], [7, [7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-well-definedness · candidate
51. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0, 2], [1, 3]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 2], [1, 3]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
52. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
53. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
54. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 1, 2, 3, 4], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 0, 0, Yes], [2, 4, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 0, 0, Yes], [3, 3, 1, 1, Yes], [3, 4, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 0, 0, Yes], [4, 2, 1, 1, Yes], [4, 3, 2, 2, Yes], [4, 4, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 1, 2, 3, 4], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 0, 0, Yes], [2, 4, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 0, 0, Yes], [3, 3, 1, 1, Yes], [3, 4, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 0, 0, Yes], [4, 2, 1, 1, Yes], [4, 3, 2, 2, Yes], [4, 4, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
55. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 2, 4, 1, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 2, 4, 1, 3], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 1, 1, Yes], [0, 4, 3, 3, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 1, 1, Yes], [1, 3, 3, 3, Yes], [1, 4, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 1, 1, Yes], [2, 2, 3, 3, Yes], [2, 3, 0, 0, Yes], [2, 4, 2, 2, Yes], [3, 0, 1, 1, Yes], [3, 1, 3, 3, Yes], [3, 2, 0, 0, Yes], [3, 3, 2, 2, Yes], [3, 4, 4, 4, Yes], [4, 0, 3, 3, Yes], [4, 1, 0, 0, Yes], [4, 2, 2, 2, Yes], [4, 3, 4, 4, Yes], [4, 4, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 2, 4, 1, 3], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 1, 1, Yes], [0, 4, 3, 3, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 1, 1, Yes], [1, 3, 3, 3, Yes], [1, 4, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 1, 1, Yes], [2, 2, 3, 3, Yes], [2, 3, 0, 0, Yes], [2, 4, 2, 2, Yes], [3, 0, 1, 1, Yes], [3, 1, 3, 3, Yes], [3, 2, 0, 0, Yes], [3, 3, 2, 2, Yes], [3, 4, 4, 4, Yes], [4, 0, 3, 3, Yes], [4, 1, 0, 0, Yes], [4, 2, 2, 2, Yes], [4, 3, 4, 4, Yes], [4, 4, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
56. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 3, 1, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 1, 1, Yes], [0, 3, 4, 4, Yes], [0, 4, 2, 2, Yes], [1, 0, 3, 3, Yes], [1, 1, 1, 1, Yes], [1, 2, 4, 4, Yes], [1, 3, 2, 2, Yes], [1, 4, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 4, 4, Yes], [2, 2, 2, 2, Yes], [2, 3, 0, 0, Yes], [2, 4, 3, 3, Yes], [3, 0, 4, 4, Yes], [3, 1, 2, 2, Yes], [3, 2, 0, 0, Yes], [3, 3, 3, 3, Yes], [3, 4, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 0, 0, Yes], [4, 2, 3, 3, Yes], [4, 3, 1, 1, Yes], [4, 4, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 3, 1, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 1, 1, Yes], [0, 3, 4, 4, Yes], [0, 4, 2, 2, Yes], [1, 0, 3, 3, Yes], [1, 1, 1, 1, Yes], [1, 2, 4, 4, Yes], [1, 3, 2, 2, Yes], [1, 4, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 4, 4, Yes], [2, 2, 2, 2, Yes], [2, 3, 0, 0, Yes], [2, 4, 3, 3, Yes], [3, 0, 4, 4, Yes], [3, 1, 2, 2, Yes], [3, 2, 0, 0, Yes], [3, 3, 3, 3, Yes], [3, 4, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 0, 0, Yes], [4, 2, 3, 3, Yes], [4, 3, 1, 1, Yes], [4, 4, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
57. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 3, 3, Yes], [0, 3, 2, 2, Yes], [0, 4, 1, 1, Yes], [1, 0, 4, 4, Yes], [1, 1, 3, 3, Yes], [1, 2, 2, 2, Yes], [1, 3, 1, 1, Yes], [1, 4, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 1, 1, Yes], [2, 3, 0, 0, Yes], [2, 4, 4, 4, Yes], [3, 0, 2, 2, Yes], [3, 1, 1, 1, Yes], [3, 2, 0, 0, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [4, 0, 1, 1, Yes], [4, 1, 0, 0, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 3, 3, Yes], [0, 3, 2, 2, Yes], [0, 4, 1, 1, Yes], [1, 0, 4, 4, Yes], [1, 1, 3, 3, Yes], [1, 2, 2, 2, Yes], [1, 3, 1, 1, Yes], [1, 4, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 1, 1, Yes], [2, 3, 0, 0, Yes], [2, 4, 4, 4, Yes], [3, 0, 2, 2, Yes], [3, 1, 1, 1, Yes], [3, 2, 0, 0, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [4, 0, 1, 1, Yes], [4, 1, 0, 0, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
58. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0, 2, 4], [1, 3, 5]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 2, 4], [1, 3, 5]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
59. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 1, 2, 0, 1, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [0, 1, 2], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 0, 0, Yes], [2, 2, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [0, 1, 2], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 0, 0, Yes], [2, 2, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
60. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 2, 1, 0, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [0, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 1, 1, Yes], [1, 0, 2, 2, Yes], [1, 1, 1, 1, Yes], [1, 2, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 0, 0, Yes], [2, 2, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [0, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 1, 1, Yes], [1, 0, 2, 2, Yes], [1, 1, 1, 1, Yes], [1, 2, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 0, 0, Yes], [2, 2, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
61. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 1, 2, 3, 4, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [0, 1, 2, 3, 4, 5], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 0, 0, Yes], [2, 5, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 0, 0, Yes], [3, 4, 1, 1, Yes], [3, 5, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 0, 0, Yes], [4, 3, 1, 1, Yes], [4, 4, 2, 2, Yes], [4, 5, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 0, 0, Yes], [5, 2, 1, 1, Yes], [5, 3, 2, 2, Yes], [5, 4, 3, 3, Yes], [5, 5, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [0, 1, 2, 3, 4, 5], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 0, 0, Yes], [2, 5, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 0, 0, Yes], [3, 4, 1, 1, Yes], [3, 5, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 0, 0, Yes], [4, 3, 1, 1, Yes], [4, 4, 2, 2, Yes], [4, 5, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 0, 0, Yes], [5, 2, 1, 1, Yes], [5, 3, 2, 2, Yes], [5, 4, 3, 3, Yes], [5, 5, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
62. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 4, 4, Yes], [0, 3, 3, 3, Yes], [0, 4, 2, 2, Yes], [0, 5, 1, 1, Yes], [1, 0, 5, 5, Yes], [1, 1, 4, 4, Yes], [1, 2, 3, 3, Yes], [1, 3, 2, 2, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 3, 3, Yes], [2, 2, 2, 2, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 5, 5, Yes], [3, 0, 3, 3, Yes], [3, 1, 2, 2, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [4, 0, 2, 2, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 4, 4, Yes], [0, 3, 3, 3, Yes], [0, 4, 2, 2, Yes], [0, 5, 1, 1, Yes], [1, 0, 5, 5, Yes], [1, 1, 4, 4, Yes], [1, 2, 3, 3, Yes], [1, 3, 2, 2, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 3, 3, Yes], [2, 2, 2, 2, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 5, 5, Yes], [3, 0, 3, 3, Yes], [3, 1, 2, 2, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [4, 0, 2, 2, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
63. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 1, 2, 3, 4, 5, 6], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 0, 0, Yes], [2, 6, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 0, 0, Yes], [3, 5, 1, 1, Yes], [3, 6, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 0, 0, Yes], [4, 4, 1, 1, Yes], [4, 5, 2, 2, Yes], [4, 6, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 0, 0, Yes], [5, 3, 1, 1, Yes], [5, 4, 2, 2, Yes], [5, 5, 3, 3, Yes], [5, 6, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 0, 0, Yes], [6, 2, 1, 1, Yes], [6, 3, 2, 2, Yes], [6, 4, 3, 3, Yes], [6, 5, 4, 4, Yes], [6, 6, 5, 5, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 1, 2, 3, 4, 5, 6], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 0, 0, Yes], [2, 6, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 0, 0, Yes], [3, 5, 1, 1, Yes], [3, 6, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 0, 0, Yes], [4, 4, 1, 1, Yes], [4, 5, 2, 2, Yes], [4, 6, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 0, 0, Yes], [5, 3, 1, 1, Yes], [5, 4, 2, 2, Yes], [5, 5, 3, 3, Yes], [5, 6, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 0, 0, Yes], [6, 2, 1, 1, Yes], [6, 3, 2, 2, Yes], [6, 4, 3, 3, Yes], [6, 5, 4, 4, Yes], [6, 6, 5, 5, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
64. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 1, 3, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 2, 4, 6, 1, 3, 5], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 6, 6, Yes], [0, 4, 1, 1, Yes], [0, 5, 3, 3, Yes], [0, 6, 5, 5, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 6, 6, Yes], [1, 3, 1, 1, Yes], [1, 4, 3, 3, Yes], [1, 5, 5, 5, Yes], [1, 6, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 6, 6, Yes], [2, 2, 1, 1, Yes], [2, 3, 3, 3, Yes], [2, 4, 5, 5, Yes], [2, 5, 0, 0, Yes], [2, 6, 2, 2, Yes], [3, 0, 6, 6, Yes], [3, 1, 1, 1, Yes], [3, 2, 3, 3, Yes], [3, 3, 5, 5, Yes], [3, 4, 0, 0, Yes], [3, 5, 2, 2, Yes], [3, 6, 4, 4, Yes], [4, 0, 1, 1, Yes], [4, 1, 3, 3, Yes], [4, 2, 5, 5, Yes], [4, 3, 0, 0, Yes], [4, 4, 2, 2, Yes], [4, 5, 4, 4, Yes], [4, 6, 6, 6, Yes], [5, 0, 3, 3, Yes], [5, 1, 5, 5, Yes], [5, 2, 0, 0, Yes], [5, 3, 2, 2, Yes], [5, 4, 4, 4, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [6, 0, 5, 5, Yes], [6, 1, 0, 0, Yes], [6, 2, 2, 2, Yes], [6, 3, 4, 4, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 2, 4, 6, 1, 3, 5], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 6, 6, Yes], [0, 4, 1, 1, Yes], [0, 5, 3, 3, Yes], [0, 6, 5, 5, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 6, 6, Yes], [1, 3, 1, 1, Yes], [1, 4, 3, 3, Yes], [1, 5, 5, 5, Yes], [1, 6, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 6, 6, Yes], [2, 2, 1, 1, Yes], [2, 3, 3, 3, Yes], [2, 4, 5, 5, Yes], [2, 5, 0, 0, Yes], [2, 6, 2, 2, Yes], [3, 0, 6, 6, Yes], [3, 1, 1, 1, Yes], [3, 2, 3, 3, Yes], [3, 3, 5, 5, Yes], [3, 4, 0, 0, Yes], [3, 5, 2, 2, Yes], [3, 6, 4, 4, Yes], [4, 0, 1, 1, Yes], [4, 1, 3, 3, Yes], [4, 2, 5, 5, Yes], [4, 3, 0, 0, Yes], [4, 4, 2, 2, Yes], [4, 5, 4, 4, Yes], [4, 6, 6, 6, Yes], [5, 0, 3, 3, Yes], [5, 1, 5, 5, Yes], [5, 2, 0, 0, Yes], [5, 3, 2, 2, Yes], [5, 4, 4, 4, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [6, 0, 5, 5, Yes], [6, 1, 0, 0, Yes], [6, 2, 2, 2, Yes], [6, 3, 4, 4, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
65. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 3, 6, 2, 5, 1, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 3, 6, 2, 5, 1, 4], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 2, 2, Yes], [0, 4, 5, 5, Yes], [0, 5, 1, 1, Yes], [0, 6, 4, 4, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 2, 2, Yes], [1, 3, 5, 5, Yes], [1, 4, 1, 1, Yes], [1, 5, 4, 4, Yes], [1, 6, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 2, 2, Yes], [2, 2, 5, 5, Yes], [2, 3, 1, 1, Yes], [2, 4, 4, 4, Yes], [2, 5, 0, 0, Yes], [2, 6, 3, 3, Yes], [3, 0, 2, 2, Yes], [3, 1, 5, 5, Yes], [3, 2, 1, 1, Yes], [3, 3, 4, 4, Yes], [3, 4, 0, 0, Yes], [3, 5, 3, 3, Yes], [3, 6, 6, 6, Yes], [4, 0, 5, 5, Yes], [4, 1, 1, 1, Yes], [4, 2, 4, 4, Yes], [4, 3, 0, 0, Yes], [4, 4, 3, 3, Yes], [4, 5, 6, 6, Yes], [4, 6, 2, 2, Yes], [5, 0, 1, 1, Yes], [5, 1, 4, 4, Yes], [5, 2, 0, 0, Yes], [5, 3, 3, 3, Yes], [5, 4, 6, 6, Yes], [5, 5, 2, 2, Yes], [5, 6, 5, 5, Yes], [6, 0, 4, 4, Yes], [6, 1, 0, 0, Yes], [6, 2, 3, 3, Yes], [6, 3, 6, 6, Yes], [6, 4, 2, 2, Yes], [6, 5, 5, 5, Yes], [6, 6, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 3, 6, 2, 5, 1, 4], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 2, 2, Yes], [0, 4, 5, 5, Yes], [0, 5, 1, 1, Yes], [0, 6, 4, 4, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 2, 2, Yes], [1, 3, 5, 5, Yes], [1, 4, 1, 1, Yes], [1, 5, 4, 4, Yes], [1, 6, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 2, 2, Yes], [2, 2, 5, 5, Yes], [2, 3, 1, 1, Yes], [2, 4, 4, 4, Yes], [2, 5, 0, 0, Yes], [2, 6, 3, 3, Yes], [3, 0, 2, 2, Yes], [3, 1, 5, 5, Yes], [3, 2, 1, 1, Yes], [3, 3, 4, 4, Yes], [3, 4, 0, 0, Yes], [3, 5, 3, 3, Yes], [3, 6, 6, 6, Yes], [4, 0, 5, 5, Yes], [4, 1, 1, 1, Yes], [4, 2, 4, 4, Yes], [4, 3, 0, 0, Yes], [4, 4, 3, 3, Yes], [4, 5, 6, 6, Yes], [4, 6, 2, 2, Yes], [5, 0, 1, 1, Yes], [5, 1, 4, 4, Yes], [5, 2, 0, 0, Yes], [5, 3, 3, 3, Yes], [5, 4, 6, 6, Yes], [5, 5, 2, 2, Yes], [5, 6, 5, 5, Yes], [6, 0, 4, 4, Yes], [6, 1, 0, 0, Yes], [6, 2, 3, 3, Yes], [6, 3, 6, 6, Yes], [6, 4, 2, 2, Yes], [6, 5, 5, 5, Yes], [6, 6, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
66. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 4, 1, 5, 2, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 4, 1, 5, 2, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 1, 1, Yes], [0, 3, 5, 5, Yes], [0, 4, 2, 2, Yes], [0, 5, 6, 6, Yes], [0, 6, 3, 3, Yes], [1, 0, 4, 4, Yes], [1, 1, 1, 1, Yes], [1, 2, 5, 5, Yes], [1, 3, 2, 2, Yes], [1, 4, 6, 6, Yes], [1, 5, 3, 3, Yes], [1, 6, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 5, 5, Yes], [2, 2, 2, 2, Yes], [2, 3, 6, 6, Yes], [2, 4, 3, 3, Yes], [2, 5, 0, 0, Yes], [2, 6, 4, 4, Yes], [3, 0, 5, 5, Yes], [3, 1, 2, 2, Yes], [3, 2, 6, 6, Yes], [3, 3, 3, 3, Yes], [3, 4, 0, 0, Yes], [3, 5, 4, 4, Yes], [3, 6, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 6, 6, Yes], [4, 2, 3, 3, Yes], [4, 3, 0, 0, Yes], [4, 4, 4, 4, Yes], [4, 5, 1, 1, Yes], [4, 6, 5, 5, Yes], [5, 0, 6, 6, Yes], [5, 1, 3, 3, Yes], [5, 2, 0, 0, Yes], [5, 3, 4, 4, Yes], [5, 4, 1, 1, Yes], [5, 5, 5, 5, Yes], [5, 6, 2, 2, Yes], [6, 0, 3, 3, Yes], [6, 1, 0, 0, Yes], [6, 2, 4, 4, Yes], [6, 3, 1, 1, Yes], [6, 4, 5, 5, Yes], [6, 5, 2, 2, Yes], [6, 6, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 4, 1, 5, 2, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 1, 1, Yes], [0, 3, 5, 5, Yes], [0, 4, 2, 2, Yes], [0, 5, 6, 6, Yes], [0, 6, 3, 3, Yes], [1, 0, 4, 4, Yes], [1, 1, 1, 1, Yes], [1, 2, 5, 5, Yes], [1, 3, 2, 2, Yes], [1, 4, 6, 6, Yes], [1, 5, 3, 3, Yes], [1, 6, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 5, 5, Yes], [2, 2, 2, 2, Yes], [2, 3, 6, 6, Yes], [2, 4, 3, 3, Yes], [2, 5, 0, 0, Yes], [2, 6, 4, 4, Yes], [3, 0, 5, 5, Yes], [3, 1, 2, 2, Yes], [3, 2, 6, 6, Yes], [3, 3, 3, 3, Yes], [3, 4, 0, 0, Yes], [3, 5, 4, 4, Yes], [3, 6, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 6, 6, Yes], [4, 2, 3, 3, Yes], [4, 3, 0, 0, Yes], [4, 4, 4, 4, Yes], [4, 5, 1, 1, Yes], [4, 6, 5, 5, Yes], [5, 0, 6, 6, Yes], [5, 1, 3, 3, Yes], [5, 2, 0, 0, Yes], [5, 3, 4, 4, Yes], [5, 4, 1, 1, Yes], [5, 5, 5, 5, Yes], [5, 6, 2, 2, Yes], [6, 0, 3, 3, Yes], [6, 1, 0, 0, Yes], [6, 2, 4, 4, Yes], [6, 3, 1, 1, Yes], [6, 4, 5, 5, Yes], [6, 5, 2, 2, Yes], [6, 6, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
67. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 5, 3, 1, 6, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 5, 3, 1, 6, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 3, 3, Yes], [0, 3, 1, 1, Yes], [0, 4, 6, 6, Yes], [0, 5, 4, 4, Yes], [0, 6, 2, 2, Yes], [1, 0, 5, 5, Yes], [1, 1, 3, 3, Yes], [1, 2, 1, 1, Yes], [1, 3, 6, 6, Yes], [1, 4, 4, 4, Yes], [1, 5, 2, 2, Yes], [1, 6, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 1, 1, Yes], [2, 2, 6, 6, Yes], [2, 3, 4, 4, Yes], [2, 4, 2, 2, Yes], [2, 5, 0, 0, Yes], [2, 6, 5, 5, Yes], [3, 0, 1, 1, Yes], [3, 1, 6, 6, Yes], [3, 2, 4, 4, Yes], [3, 3, 2, 2, Yes], [3, 4, 0, 0, Yes], [3, 5, 5, 5, Yes], [3, 6, 3, 3, Yes], [4, 0, 6, 6, Yes], [4, 1, 4, 4, Yes], [4, 2, 2, 2, Yes], [4, 3, 0, 0, Yes], [4, 4, 5, 5, Yes], [4, 5, 3, 3, Yes], [4, 6, 1, 1, Yes], [5, 0, 4, 4, Yes], [5, 1, 2, 2, Yes], [5, 2, 0, 0, Yes], [5, 3, 5, 5, Yes], [5, 4, 3, 3, Yes], [5, 5, 1, 1, Yes], [5, 6, 6, 6, Yes], [6, 0, 2, 2, Yes], [6, 1, 0, 0, Yes], [6, 2, 5, 5, Yes], [6, 3, 3, 3, Yes], [6, 4, 1, 1, Yes], [6, 5, 6, 6, Yes], [6, 6, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 5, 3, 1, 6, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 3, 3, Yes], [0, 3, 1, 1, Yes], [0, 4, 6, 6, Yes], [0, 5, 4, 4, Yes], [0, 6, 2, 2, Yes], [1, 0, 5, 5, Yes], [1, 1, 3, 3, Yes], [1, 2, 1, 1, Yes], [1, 3, 6, 6, Yes], [1, 4, 4, 4, Yes], [1, 5, 2, 2, Yes], [1, 6, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 1, 1, Yes], [2, 2, 6, 6, Yes], [2, 3, 4, 4, Yes], [2, 4, 2, 2, Yes], [2, 5, 0, 0, Yes], [2, 6, 5, 5, Yes], [3, 0, 1, 1, Yes], [3, 1, 6, 6, Yes], [3, 2, 4, 4, Yes], [3, 3, 2, 2, Yes], [3, 4, 0, 0, Yes], [3, 5, 5, 5, Yes], [3, 6, 3, 3, Yes], [4, 0, 6, 6, Yes], [4, 1, 4, 4, Yes], [4, 2, 2, 2, Yes], [4, 3, 0, 0, Yes], [4, 4, 5, 5, Yes], [4, 5, 3, 3, Yes], [4, 6, 1, 1, Yes], [5, 0, 4, 4, Yes], [5, 1, 2, 2, Yes], [5, 2, 0, 0, Yes], [5, 3, 5, 5, Yes], [5, 4, 3, 3, Yes], [5, 5, 1, 1, Yes], [5, 6, 6, 6, Yes], [6, 0, 2, 2, Yes], [6, 1, 0, 0, Yes], [6, 2, 5, 5, Yes], [6, 3, 3, 3, Yes], [6, 4, 1, 1, Yes], [6, 5, 6, 6, Yes], [6, 6, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
68. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, and the aligned total mapping=[0, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 6, 6, Yes], [0, 2, 5, 5, Yes], [0, 3, 4, 4, Yes], [0, 4, 3, 3, Yes], [0, 5, 2, 2, Yes], [0, 6, 1, 1, Yes], [1, 0, 6, 6, Yes], [1, 1, 5, 5, Yes], [1, 2, 4, 4, Yes], [1, 3, 3, 3, Yes], [1, 4, 2, 2, Yes], [1, 5, 1, 1, Yes], [1, 6, 0, 0, Yes], [2, 0, 5, 5, Yes], [2, 1, 4, 4, Yes], [2, 2, 3, 3, Yes], [2, 3, 2, 2, Yes], [2, 4, 1, 1, Yes], [2, 5, 0, 0, Yes], [2, 6, 6, 6, Yes], [3, 0, 4, 4, Yes], [3, 1, 3, 3, Yes], [3, 2, 2, 2, Yes], [3, 3, 1, 1, Yes], [3, 4, 0, 0, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [4, 0, 3, 3, Yes], [4, 1, 2, 2, Yes], [4, 2, 1, 1, Yes], [4, 3, 0, 0, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [5, 0, 2, 2, Yes], [5, 1, 1, 1, Yes], [5, 2, 0, 0, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [6, 0, 1, 1, Yes], [6, 1, 0, 0, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 6, 6, Yes], [0, 2, 5, 5, Yes], [0, 3, 4, 4, Yes], [0, 4, 3, 3, Yes], [0, 5, 2, 2, Yes], [0, 6, 1, 1, Yes], [1, 0, 6, 6, Yes], [1, 1, 5, 5, Yes], [1, 2, 4, 4, Yes], [1, 3, 3, 3, Yes], [1, 4, 2, 2, Yes], [1, 5, 1, 1, Yes], [1, 6, 0, 0, Yes], [2, 0, 5, 5, Yes], [2, 1, 4, 4, Yes], [2, 2, 3, 3, Yes], [2, 3, 2, 2, Yes], [2, 4, 1, 1, Yes], [2, 5, 0, 0, Yes], [2, 6, 6, 6, Yes], [3, 0, 4, 4, Yes], [3, 1, 3, 3, Yes], [3, 2, 2, 2, Yes], [3, 3, 1, 1, Yes], [3, 4, 0, 0, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [4, 0, 3, 3, Yes], [4, 1, 2, 2, Yes], [4, 2, 1, 1, Yes], [4, 3, 0, 0, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [5, 0, 2, 2, Yes], [5, 1, 1, 1, Yes], [5, 2, 0, 0, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [6, 0, 1, 1, Yes], [6, 1, 0, 0, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
69. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0, 2, 4, 6], [1, 3, 5, 7]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 2, 4, 6], [1, 3, 5, 7]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
70. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
71. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1, 0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
72. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 1, 2, 3, 4, 5, 6, 7], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [0, 7, 7, 7, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 7, 7, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 7, 7, Yes], [2, 6, 0, 0, Yes], [2, 7, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 7, 7, Yes], [3, 5, 0, 0, Yes], [3, 6, 1, 1, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 7, 7, Yes], [4, 4, 0, 0, Yes], [4, 5, 1, 1, Yes], [4, 6, 2, 2, Yes], [4, 7, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 7, 7, Yes], [5, 3, 0, 0, Yes], [5, 4, 1, 1, Yes], [5, 5, 2, 2, Yes], [5, 6, 3, 3, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 7, 7, Yes], [6, 2, 0, 0, Yes], [6, 3, 1, 1, Yes], [6, 4, 2, 2, Yes], [6, 5, 3, 3, Yes], [6, 6, 4, 4, Yes], [6, 7, 5, 5, Yes], [7, 0, 7, 7, Yes], [7, 1, 0, 0, Yes], [7, 2, 1, 1, Yes], [7, 3, 2, 2, Yes], [7, 4, 3, 3, Yes], [7, 5, 4, 4, Yes], [7, 6, 5, 5, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 1, 2, 3, 4, 5, 6, 7], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [0, 7, 7, 7, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 7, 7, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 7, 7, Yes], [2, 6, 0, 0, Yes], [2, 7, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 7, 7, Yes], [3, 5, 0, 0, Yes], [3, 6, 1, 1, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 7, 7, Yes], [4, 4, 0, 0, Yes], [4, 5, 1, 1, Yes], [4, 6, 2, 2, Yes], [4, 7, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 7, 7, Yes], [5, 3, 0, 0, Yes], [5, 4, 1, 1, Yes], [5, 5, 2, 2, Yes], [5, 6, 3, 3, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 7, 7, Yes], [6, 2, 0, 0, Yes], [6, 3, 1, 1, Yes], [6, 4, 2, 2, Yes], [6, 5, 3, 3, Yes], [6, 6, 4, 4, Yes], [6, 7, 5, 5, Yes], [7, 0, 7, 7, Yes], [7, 1, 0, 0, Yes], [7, 2, 1, 1, Yes], [7, 3, 2, 2, Yes], [7, 4, 3, 3, Yes], [7, 5, 4, 4, Yes], [7, 6, 5, 5, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
73. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 3, 6, 1, 4, 7, 2, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 3, 6, 1, 4, 7, 2, 5], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 1, 1, Yes], [0, 4, 4, 4, Yes], [0, 5, 7, 7, Yes], [0, 6, 2, 2, Yes], [0, 7, 5, 5, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 1, 1, Yes], [1, 3, 4, 4, Yes], [1, 4, 7, 7, Yes], [1, 5, 2, 2, Yes], [1, 6, 5, 5, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 1, 1, Yes], [2, 2, 4, 4, Yes], [2, 3, 7, 7, Yes], [2, 4, 2, 2, Yes], [2, 5, 5, 5, Yes], [2, 6, 0, 0, Yes], [2, 7, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 4, 4, Yes], [3, 2, 7, 7, Yes], [3, 3, 2, 2, Yes], [3, 4, 5, 5, Yes], [3, 5, 0, 0, Yes], [3, 6, 3, 3, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 7, 7, Yes], [4, 2, 2, 2, Yes], [4, 3, 5, 5, Yes], [4, 4, 0, 0, Yes], [4, 5, 3, 3, Yes], [4, 6, 6, 6, Yes], [4, 7, 1, 1, Yes], [5, 0, 7, 7, Yes], [5, 1, 2, 2, Yes], [5, 2, 5, 5, Yes], [5, 3, 0, 0, Yes], [5, 4, 3, 3, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 5, 5, Yes], [6, 2, 0, 0, Yes], [6, 3, 3, 3, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 4, 4, Yes], [6, 7, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 0, 0, Yes], [7, 2, 3, 3, Yes], [7, 3, 6, 6, Yes], [7, 4, 1, 1, Yes], [7, 5, 4, 4, Yes], [7, 6, 7, 7, Yes], [7, 7, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 3, 6, 1, 4, 7, 2, 5], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 1, 1, Yes], [0, 4, 4, 4, Yes], [0, 5, 7, 7, Yes], [0, 6, 2, 2, Yes], [0, 7, 5, 5, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 1, 1, Yes], [1, 3, 4, 4, Yes], [1, 4, 7, 7, Yes], [1, 5, 2, 2, Yes], [1, 6, 5, 5, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 1, 1, Yes], [2, 2, 4, 4, Yes], [2, 3, 7, 7, Yes], [2, 4, 2, 2, Yes], [2, 5, 5, 5, Yes], [2, 6, 0, 0, Yes], [2, 7, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 4, 4, Yes], [3, 2, 7, 7, Yes], [3, 3, 2, 2, Yes], [3, 4, 5, 5, Yes], [3, 5, 0, 0, Yes], [3, 6, 3, 3, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 7, 7, Yes], [4, 2, 2, 2, Yes], [4, 3, 5, 5, Yes], [4, 4, 0, 0, Yes], [4, 5, 3, 3, Yes], [4, 6, 6, 6, Yes], [4, 7, 1, 1, Yes], [5, 0, 7, 7, Yes], [5, 1, 2, 2, Yes], [5, 2, 5, 5, Yes], [5, 3, 0, 0, Yes], [5, 4, 3, 3, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 5, 5, Yes], [6, 2, 0, 0, Yes], [6, 3, 3, 3, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 4, 4, Yes], [6, 7, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 0, 0, Yes], [7, 2, 3, 3, Yes], [7, 3, 6, 6, Yes], [7, 4, 1, 1, Yes], [7, 5, 4, 4, Yes], [7, 6, 7, 7, Yes], [7, 7, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
74. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 5, 2, 7, 4, 1, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 5, 2, 7, 4, 1, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 2, 2, Yes], [0, 3, 7, 7, Yes], [0, 4, 4, 4, Yes], [0, 5, 1, 1, Yes], [0, 6, 6, 6, Yes], [0, 7, 3, 3, Yes], [1, 0, 5, 5, Yes], [1, 1, 2, 2, Yes], [1, 2, 7, 7, Yes], [1, 3, 4, 4, Yes], [1, 4, 1, 1, Yes], [1, 5, 6, 6, Yes], [1, 6, 3, 3, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 7, 7, Yes], [2, 2, 4, 4, Yes], [2, 3, 1, 1, Yes], [2, 4, 6, 6, Yes], [2, 5, 3, 3, Yes], [2, 6, 0, 0, Yes], [2, 7, 5, 5, Yes], [3, 0, 7, 7, Yes], [3, 1, 4, 4, Yes], [3, 2, 1, 1, Yes], [3, 3, 6, 6, Yes], [3, 4, 3, 3, Yes], [3, 5, 0, 0, Yes], [3, 6, 5, 5, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 1, 1, Yes], [4, 2, 6, 6, Yes], [4, 3, 3, 3, Yes], [4, 4, 0, 0, Yes], [4, 5, 5, 5, Yes], [4, 6, 2, 2, Yes], [4, 7, 7, 7, Yes], [5, 0, 1, 1, Yes], [5, 1, 6, 6, Yes], [5, 2, 3, 3, Yes], [5, 3, 0, 0, Yes], [5, 4, 5, 5, Yes], [5, 5, 2, 2, Yes], [5, 6, 7, 7, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 3, 3, Yes], [6, 2, 0, 0, Yes], [6, 3, 5, 5, Yes], [6, 4, 2, 2, Yes], [6, 5, 7, 7, Yes], [6, 6, 4, 4, Yes], [6, 7, 1, 1, Yes], [7, 0, 3, 3, Yes], [7, 1, 0, 0, Yes], [7, 2, 5, 5, Yes], [7, 3, 2, 2, Yes], [7, 4, 7, 7, Yes], [7, 5, 4, 4, Yes], [7, 6, 1, 1, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 5, 2, 7, 4, 1, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 2, 2, Yes], [0, 3, 7, 7, Yes], [0, 4, 4, 4, Yes], [0, 5, 1, 1, Yes], [0, 6, 6, 6, Yes], [0, 7, 3, 3, Yes], [1, 0, 5, 5, Yes], [1, 1, 2, 2, Yes], [1, 2, 7, 7, Yes], [1, 3, 4, 4, Yes], [1, 4, 1, 1, Yes], [1, 5, 6, 6, Yes], [1, 6, 3, 3, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 7, 7, Yes], [2, 2, 4, 4, Yes], [2, 3, 1, 1, Yes], [2, 4, 6, 6, Yes], [2, 5, 3, 3, Yes], [2, 6, 0, 0, Yes], [2, 7, 5, 5, Yes], [3, 0, 7, 7, Yes], [3, 1, 4, 4, Yes], [3, 2, 1, 1, Yes], [3, 3, 6, 6, Yes], [3, 4, 3, 3, Yes], [3, 5, 0, 0, Yes], [3, 6, 5, 5, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 1, 1, Yes], [4, 2, 6, 6, Yes], [4, 3, 3, 3, Yes], [4, 4, 0, 0, Yes], [4, 5, 5, 5, Yes], [4, 6, 2, 2, Yes], [4, 7, 7, 7, Yes], [5, 0, 1, 1, Yes], [5, 1, 6, 6, Yes], [5, 2, 3, 3, Yes], [5, 3, 0, 0, Yes], [5, 4, 5, 5, Yes], [5, 5, 2, 2, Yes], [5, 6, 7, 7, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 3, 3, Yes], [6, 2, 0, 0, Yes], [6, 3, 5, 5, Yes], [6, 4, 2, 2, Yes], [6, 5, 7, 7, Yes], [6, 6, 4, 4, Yes], [6, 7, 1, 1, Yes], [7, 0, 3, 3, Yes], [7, 1, 0, 0, Yes], [7, 2, 5, 5, Yes], [7, 3, 2, 2, Yes], [7, 4, 7, 7, Yes], [7, 5, 4, 4, Yes], [7, 6, 1, 1, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
75. The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).
View answer and explanation
([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 7, 7, Yes], [0, 2, 6, 6, Yes], [0, 3, 5, 5, Yes], [0, 4, 4, 4, Yes], [0, 5, 3, 3, Yes], [0, 6, 2, 2, Yes], [0, 7, 1, 1, Yes], [1, 0, 7, 7, Yes], [1, 1, 6, 6, Yes], [1, 2, 5, 5, Yes], [1, 3, 4, 4, Yes], [1, 4, 3, 3, Yes], [1, 5, 2, 2, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 5, 5, Yes], [2, 2, 4, 4, Yes], [2, 3, 3, 3, Yes], [2, 4, 2, 2, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 7, 7, Yes], [3, 0, 5, 5, Yes], [3, 1, 4, 4, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 3, 3, Yes], [4, 2, 2, 2, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [5, 0, 3, 3, Yes], [5, 1, 2, 2, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)
Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 7, 7, Yes], [0, 2, 6, 6, Yes], [0, 3, 5, 5, Yes], [0, 4, 4, 4, Yes], [0, 5, 3, 3, Yes], [0, 6, 2, 2, Yes], [0, 7, 1, 1, Yes], [1, 0, 7, 7, Yes], [1, 1, 6, 6, Yes], [1, 2, 5, 5, Yes], [1, 3, 4, 4, Yes], [1, 4, 3, 3, Yes], [1, 5, 2, 2, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 5, 5, Yes], [2, 2, 4, 4, Yes], [2, 3, 3, 3, Yes], [2, 4, 2, 2, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 7, 7, Yes], [3, 0, 5, 5, Yes], [3, 1, 4, 4, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 3, 3, Yes], [4, 2, 2, 2, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [5, 0, 3, 3, Yes], [5, 1, 2, 2, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-induced-isomorphism-audit · candidate
76. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1], [1, 0]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2], 4, 2, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2], 4, 2, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
77. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, target 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, and the aligned total mapping=[0, 2, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4], 6, 3, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4], 6, 3, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
78. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target 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, and the aligned total mapping=[0, 2, 4, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6], 8, 4, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6], 8, 4, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
79. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target 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, and the aligned total mapping=[0, 2, 4, 6, 8]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8], 10, 5, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8], 10, 5, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
80. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source 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]], identity=0, target 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, and the aligned total mapping=[0, 2, 4, 6, 8, 10]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10], 12, 6, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], 12, 6, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
81. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], identity=0, target 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, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12], 14, 7, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12], 14, 7, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
82. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14], 16, 8, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14], 16, 8, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
83. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16], 18, 9, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16], 18, 9, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
84. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 20, 10, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 20, 10, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
85. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 22, 11, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 22, 11, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
86. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 24, 12, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 24, 12, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
87. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
88. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source 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]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 28, 14, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 28, 14, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
89. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 30, 15, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 30, 15, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
90. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 32, 16, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 32, 16, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
91. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 34, 17, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 34, 17, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
92. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 36, 18, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 36, 18, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
93. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 38, 19, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 38, 19, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
94. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
95. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 42, 21, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 42, 21, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
96. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 44, 22, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 44, 22, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
97. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 46, 23, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 46, 23, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
98. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 48, 24, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 48, 24, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
99. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 50, 25, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 50, 25, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
100. First Isomorphism Reaches the Image, Not Automatically the Whole Codomain
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).
The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism.
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 52, 26, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)
Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 52, 26, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof).
Reviewed step-by-step derivation
First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate
Canonical question JSONL · English JSONL · Validation report