# Conjugation

From Groupprops

## Definition

Let be a group and be an element. Then, the **conjugation map** by , denoted , is defined as the map:

.

In other words, .

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

and further, this is denoted as .

## Facts

- The conjugation map by any 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 for some .
- Conjugate elements: Two elements are termed conjugate if there exists such that .
- Conjugacy class: The conjugacy class of is the set of all elements that can be written as for some .