Conjugation
From Groupprops
Definition
Let G be a group and
be an element. Then, the conjugation map by g, denoted cg, is defined as the map:
.
In other words, cg(x) = gxg − 1.
Note that when the convention is to make the group act on the right, conjugation by g is defined as:
and further, this is denoted as xg.
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 cg for some
.
- Conjugate elements: Two elements
are termed conjugate if there exists
such that gxg − 1 = y.
- Conjugacy class: The conjugacy class of
is the set of all elements that can be written as gxg − 1 for some
.