# Finite normal subgroup

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

A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset.

mko-finite-normal-subgroup · definition · v0.10.0



## Mathematical statement

The complete Cayley table must certify G, and H must first pass identity, product and inverse closure. Normality then requires gHg^(-1)=H for every g in G, equivalently gH=Hg for every g. One explicit g,h with h in H but g*h*g^(-1) outside H refutes normality. Trivial and whole subgroups are normal. Finiteness, sampled conjugates or generator checks without a proved generation obligation are insufficient. Well-defined quotient multiplication is not yet claimed here.





## Representative formula (exact TeX)

N(H,G)=1





## Assumptions


- Families use complete finite Cayley tables, an identity and an explicit subset, not sampled multiplication.
- Conjugate sets, left/right cosets and inverses are completely enumerated from the same table.





## Symbol scope


- N — normality predicate
- H — candidate subgroup
- G — finite group





## Intuition, definition, and misconceptions

Subgroup and normal-subgroup qualification are different obligations.

Full conjugate-set equality and all-g left/right coset equality are equivalent checks.

One escaping conjugate element refutes normality.

Neither finiteness nor a subset size dividing |G| makes it normal; a few sampled g values are not enough.





## Adopted derivation guide

[Normality of rotations from every conjugate](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-normal-subgroup-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/) — Use the complete Cayley table.
- [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/) — Subgroup closure comes first.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Normality has an all-g obligation.





## Verification boundary

Only complete finite group models are handled; quotient operations and the first isomorphism theorem are not materialized here.

not_formalized

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





## Next obligation

Add finite quotient groups, kernels and the first isomorphism theorem.





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