Conjugation

Overview

Let $G$ be a group. The automorphism $i_{g} : G \to G$ , where $i_{g} (x) = g x g^{- 1}$ for all $x \in G$ , is the inner automorphism of $G$ by $g$ . Performing $i_{g}$ on $x$ is called conjugation of $x$ by $g$ . We say two elements $a, b \in G$ are conjugate if there exists some $g \in G$ such that $a = g b g^{- 1}$ .

The normal subgroups of a group $G$ are precisely those that are invariant under all inner automorphisms. A subgroup $K$ of $G$ is a conjugate subgroup of $H$ if $K = i_{g} [[H]]$ for some $g \in G$ .

The set of inner automorphisms of $G$ under the induced operation forms a normal subgroup of $Aut (G)$ , the automorphism group of $G$ . This group is denoted as $Inn (G)$ . The quotient group $Aut (G) / Inn (G)$ is the outer automorphism group, denoted as $Out (G)$ .

Equivalence Relations

Conjugacy Classes

Let $G$ be a group and $a, b \in G$ . We define relation $\sim$ as follows:

a \sim b ⟺ \exists g \in G, a = g b g^{- 1} .

Then $\sim$ is an equivalence relation. The equivalence class containing $a \in G$ is called the conjugacy class of $a$ , denoted $Cl (a)$ .

By Subgroup

Let $G$ be a group and $H, K \leq G$ . Let $i_{g}$ denote the inner automorphism of $G$ by $g \in G$ . We define relation $\sim$ as follows:

H \sim K ⟺ \exists g \in G, i_{g} [[H]] = K .

Then $\sim$ is an equivalence relation. The normal subgroups of $G$ are precisely the subgroups in the one-element equivalence classes.