# First isomorphism theorem for finite groups

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain.

mko-first-isomorphism-theorem-finite-groups · theorem · v0.10.0



## Mathematical statement

Qualify both complete Cayley tables and the complete total homomorphism f first. Construct the full kernel, image and kernel-coset quotient. The induced map bar_f(g*kernel)=f(g) must agree on every representative of every coset, preserve every product and be injective and surjective onto the image. Equal cardinalities alone do not prove an isomorphism. If f is not surjective, do not replace the image by the entire codomain. Finite enumeration certifies the specified model, not the general abstract theorem; a nonhomomorphism cannot invoke the theorem.





## Representative formula (exact TeX)

I(f)=h





## Assumptions


- Families use two complete finite group tables and an aligned total mapping array.
- The quotient and image carriers are constructed by complete finite enumeration.





## Symbol scope


- I — first-isomorphism construction
- f — finite group homomorphism
- h — induced group isomorphism





## Intuition, definition, and misconceptions

Qualify the homomorphism, kernel, image and quotient first.

Representative independence precedes the induced-map conclusion.

Check the full homomorphism and bijection, retaining the image/codomain boundary.

Equal cardinalities alone do not certify isomorphism, and a nonsurjective f does not identify G/kernel with the entire target group.





## Adopted derivation guide

[The correct image isomorphism for a nonsurjection](https://ocme.evemisslab.com/en/derivations/derivation-mko-first-isomorphism-theorem-finite-groups-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Kernel of a finite group homomorphism](https://ocme.evemisslab.com/en/mko/mko-finite-group-homomorphism-kernel/) — Use kernel and fiber cosets.
- [Finite quotient group](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-group/) — Require the full quotient carrier and well-defined multiplication.
- [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/) — The image forms a subgroup of the target.
- [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/mko/mko-bijective/) — Check injectivity and surjectivity of the induced map.





## Verification boundary

These complete finite-enumeration certificates do not replace the general abstract proof of first isomorphism.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add group-center and class-equation structural consequences.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/first-isomorphism-theorem-finite-groups.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-first-isomorphism-theorem-finite-groups/index.md)