# Center of a finite group

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

The center consists of all elements commuting with every group element and is a normal subgroup.

mko-finite-group-center · definition · v0.10.0



## Mathematical statement

From a complete qualified Cayley table, build each element z's commutation row against every g. The center Z(G) consists exactly of all-true rows. Directly check identity, product and inverse closure, then every conjugation to certify normality. z is central exactly when its full conjugacy class is a singleton, and Z(G)=G exactly when G is abelian. The center is not the centralizer of one chosen element. Characteristic status and the class equation are not asserted in this slice.





## Representative formula (exact TeX)

Z(G)=C





## Assumptions


- The complete finite Cayley table, identity and all inverses are qualified first.
- Commutation, subgroup and normality checks enumerate the entire relevant finite domain.





## Symbol scope


- Z — group-center operator
- G — finite group
- C — central subgroup





## Intuition, definition, and misconceptions

The complete commutation table defines the center.

Subgroup and normality obligations are audited separately.

Separate singleton classes, abelianness and noncentral witnesses.

Z(G) is not the same as a single-element centralizer; checking a sample of g values does not establish centrality.





## Adopted derivation guide

[Certifying the center from all commutation rows](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-group-center-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 group operation.
- [Conjugacy class in a finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group-conjugacy-class/) — Relate centrality to singleton classes.
- [Finite normal subgroup](https://ocme.evemisslab.com/en/mko/mko-finite-normal-subgroup/) — Audit center normality.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Commutation is required for every g.





## Verification boundary

No characteristic-center claim or class-equation argument is supplied 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 the class equation and p-group center results.





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