# Conjugacy class in a finite group

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

A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group.

mko-finite-group-conjugacy-class · definition · v0.10.0



## Mathematical statement

Derive the full conjugation table from the complete group table and inverses, checking identity and action composition. The class of x contains every g*x*g^(-1), and all classes are equal or disjoint and cover G. A singleton class means x commutes with every group element; an explicit y outside the class proves nonconjugacy. A conjugacy class is neither a coset nor a conjugate subgroup. No centralizer, class-size formula or additional orbit-stabilizer consequence is used here.





## Representative formula (exact TeX)

C(G,x)=O





## Assumptions


- The complete finite group table, identity and every inverse qualify first.
- Action tables, orbits and the partition are completely enumerated.





## Symbol scope


- C — conjugacy-class operator
- G — finite group
- x — group element
- O — conjugation orbit





## Intuition, definition, and misconceptions

Derive conjugation completely from the group operation.

Each class is the whole orbit of its specified element.

Singleton/nonconjugacy witnesses introduce no centralizer count.

Do not confuse a class with a left coset or a conjugate subgroup gHg^(-1).





## Adopted derivation guide

[Full conjugation action and three classes](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-group-conjugacy-class-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 action](https://ocme.evemisslab.com/en/mko/mko-finite-group-action/) — Conjugation must satisfy action axioms.
- [Orbit of a finite group action](https://ocme.evemisslab.com/en/mko/mko-finite-group-orbit/) — Orbits partition the carrier.
- [Finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) — Multiplication and inverses define this action.





## Verification boundary

Class-size formulas, the group center and the class equation are reserved for later nodes.

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 center of a finite group and the class equation.





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