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. Further information: Group acts as automorphisms by conjugation

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 $g\in G$.