Conjugation is the process of taking a complex conjugate of a complex number, complex matrix, etc., or of performing a conjugation move on a knot.
Conjugation also has a meaning in group theory. Let 
be a group
and let 
.
Then, 
defines a homomorphism 
given by
 |
This is a homomorphism because
 |
The operation on 
given by 
is called conjugation by 
.
Conjugation is an important construction in group theory. Conjugation defines a group action of a group on itself and this often yields useful information about the group. For example, this technique is how the Sylow Theorems are proven. More importantly, a normal subgroup of a group is a subgroup which is invariant under conjugation by any element. Normal groups are extremely important because they are the kernels of homomorphisms and it is possible to take the quotient of a group and one of its normal subgroups.
