Center of a finite group

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite group — Use the complete group operation.
• Conjugacy class in a finite group — Relate centrality to singleton classes.
• Finite normal subgroup — Audit center normality.
• Quantifiers and explicit domains — 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.