# Conjugation

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## 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 $g \in G$.