# Kernel of a finite group homomorphism

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

For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets.

mko-finite-group-homomorphism-kernel · definition · v0.10.0



## Mathematical statement

Both groups require complete Cayley-table qualification. f is a total mapping and must satisfy f(x*y)=f(x)*f(y) for every pair. Its kernel is the entire target-identity preimage, with subgroup and all-g conjugation closure checked directly. f(a)=f(b) exactly when a^(-1)*b belongs to the kernel, so each nonempty fiber is a kernel coset. The kernel is the source-identity singleton exactly when f is injective. Nonhomomorphisms cannot invoke these theorems; quotient/image isomorphism is not claimed here.





## Representative formula (exact TeX)

K(f)=H





## Assumptions


- Families declare two complete finite group tables and a total mapping array.
- Kernel, fibers, normality and injectivity are completely enumerated in the finite model.





## Symbol scope


- K — kernel operator
- f — finite group homomorphism
- H — normal kernel subgroup





## Intuition, definition, and misconceptions

Homomorphism qualification precedes kernel conclusions.

Audit kernel normality with complete conjugation.

Fiber/coset equivalence and trivial-kernel injectivity are separate checks.

The identity preimage of an arbitrary total map is not automatically a normal kernel, and G/kernel-image isomorphism is not established merely by naming the kernel.





## Adopted derivation guide

[Kernel, fibers and injectivity from C4 to C2](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-group-homomorphism-kernel-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/) — Use the homomorphism law and kernel background.
- [Finite normal subgroup](https://ocme.evemisslab.com/en/mko/mko-finite-normal-subgroup/) — Kernel normality has its own obligation.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The map is complete and total.





## Verification boundary

This slice focuses on normality, fibers and injectivity rather than repeating generic kernel/image exercises; the first isomorphism theorem remains later work.

not_formalized

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





## Next obligation

Add the first isomorphism theorem for finite groups.





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