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 .
- 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