Conjugation

Definition

Let $G$ be a group and $g \in G$ be an element. Then, the conjugation map by $g$ , denoted $c g$ , is defined as the map:

$x \mapsto gxg^{-1}$ .

In other words, $c g (x) = g x g - 1$ .

Note that when the convention is to make the group act on the right, conjugation by $g$ is defined as:

$x \mapsto g^{-1}xg$

and further, this is denoted as $x g$ .

The conjugation map by any $g \in G$ is an automorphism of the group; an automorphism arising this way is termed an inner automorphism.
The conjugation map defines an action of the group on itself via automorphism. Further information: Group acts as automorphisms by conjugation

Inner automorphism: An automorphism that can be expressed as $c g$ for some $g \in G$ .
Conjugate elements: Two elements $x,y \in G$ are termed conjugate if there exists $g \in G$ such that $g x g - 1 = y$ .
Conjugacy class: The conjugacy class of $x \in G$ is the set of all elements that can be written as $g x g - 1$ for some $g \in G$ .