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, $$c_g(x) = gxg^{-1}$$.

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 $$x^g$$.

Facts

 * 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.

Related terms

 * 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 $$gxg^{-1} = y$$.
 * Conjugacy class: The conjugacy class of $$x \in G$$ is the set of all elements that can be written as $$gxg^{-1}$$ for some $$g \in G$$.