# Subgroups and group homomorphisms

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

A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table.

mko-subgroup-homomorphism · concept · v0.10.0



## Mathematical statement

A nonempty subset H of a finite group G is a subgroup if it is closed under the operation, contains the identity and is closed under inverses. A homomorphism f:G→K satisfies f(a*b)=f(a)*f(b). A homomorphism is not automatically injective.





## Representative formula (exact TeX)

H(G)





## Assumptions


- Exercises provide complete finite Cayley tables for source and target and a complete mapping table.
- Conclusions concern only the specified finite groups and do not extend to arbitrary groups or make every homomorphism an isomorphism.





## Symbol scope


- H — subgroup/homomorphism record
- G — finite group





## Intuition, definition, and misconceptions

A finite subgroup audit can explicitly check closure, the identity and inverses.

The homomorphism obligation must be checked on every input pair.

The kernel consists of source elements mapped to the target identity; the image consists of all mapping values. Injectivity requires image size equal to source size.

Do not report a full subgroup-axiom audit after checking only closure or assume every homomorphism is injective or an isomorphism.





## Adopted derivation guide

[Subgroups and homomorphisms: auditing on the complete tables of ℤ/4 and ℤ/2](https://ocme.evemisslab.com/en/derivations/derivation-mko-subgroup-homomorphism-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) — Subgroups and homomorphisms presuppose already-qualified group structures.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — A subgroup carrier is a subset of the group carrier.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — A homomorphism is a total operation-preserving map.





## Verification boundary

Complete finite tables establish only the specified model.

not_formalized

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





## Next obligation

Add cosets, normal subgroups, quotient groups and isomorphism theorems.





## 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/subgroup-homomorphism.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/index.md)