Conjugacy class in a finite group

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

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 — Conjugation must satisfy action axioms.
• Orbit of a finite group action — Orbits partition the carrier.
• 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.