Conjugation by an element

In a group $𝐺$ , given elements $𝑥, 𝑔 \in 𝐺$ , we may conjugate $𝑥$ by $𝑔$ to get $𝑔 𝑎 𝑔 - 1$ . Sometimes this is written as $ˆ 𝑏 𝑎$ or $𝑏 𝑎$ , or for the right-action variant, $𝑎 𝑏 = 𝑏 - 1 𝑎 𝑏$ .

Conjugation as an action

Conjugation by a given element is an automorphism of the group, such that $̂ \cdot : 𝐺 \to A u t (𝐺)$ constitutes a group action. The orbit of an element $𝑥 \in 𝐺$ is its Conjugacy class $[𝑥] \sim$ , while its Stabilizer group is its centralizer group. An automorphism given by conjugation is called an inner automorphism, and the image $̂ 𝐺 = I n n (𝐺) ⊴ A u t (𝐺)$ forms the inner automorphism group.

Conjugacy relation

Given two group elements $𝑥, 𝑦 \in 𝐺$ , we say $𝑥$ is conjugate to $𝑦$ ( $𝑥 \sim 𝑦$ ) iff there exists $𝑔 \in 𝐺$ such that $𝑦 = 𝑔 𝑥 𝑔 - 1$ . #m/def/group The conjugacy relation $\sim$ is an Equivalence relation. #m/thm/group

Proof of equivalence relation

For any $𝑥 \in 𝐺$ , $𝑥 = 𝑒 𝑥 𝑒 - 1 ⟹ 𝑥 \sim 𝑥$ , thus $\sim$ is reflexive. For any $𝑥, 𝑦 \in 𝐺$ , $𝑥 \sim 𝑦 ⟺ 𝑦 = 𝑔 𝑥 𝑔 - 1 ⟺ 𝑥 = (𝑔 - 1) 𝑦 (𝑔 - 1) - 1 ⟺ 𝑦 \sim 𝑥$ , thus $\sim$ is symmetric. For any $𝑥, 𝑦, 𝑧 \in 𝐺$ such that $𝑥 \sim 𝑦$ and $𝑦 \sim 𝑧$ , there exist $𝑔, ℎ \in 𝐺$ such that $𝑦 = 𝑔 𝑥 𝑔 - 1$ and $𝑧 = ℎ 𝑦 ℎ - 1$ . Then $𝑧 = ℎ 𝑔 𝑥 𝑔 - 1 ℎ - 1 = (ℎ 𝑔) 𝑥 (ℎ 𝑔) - 1$ and hence $𝑥 \sim 𝑧$ , wherefore $\sim$ is transitive. Therefore $\sim$ is an equivalence relation.

A conjugacy relation may also be applied between subgroups, see Conjugate subgroups.

Conjugacy class

The equivalence classes for the conjugacy relation form so-called conjugacy classes.

[𝑥] \sim = {𝑔 𝑥 𝑔 - 1 : 𝑔 \in 𝐺}

Properties

Examples

In $S O (3)$ rotations by the same angle (i.e. only differing in axis of rotation) form conjugacy classes.
In $G L (𝑛)$ elements are conjugate to each other iff they have similar matrices (in subgroups, such as $S O (𝑛)$ , conjugacy may be more restricted, however all conjugate elements are similar).

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