# Qualified Quotient-to-Image Isomorphisms and Codomain Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0100/)

First isomorphism theorem for finite groups · qf-mod46-iso1-image-codomain-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/validation-report.json)