Conjugation

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.

REFERENCES:

Fraleigh, J. B. A First Course in Abstract Algebra, 7th ed. Reading, MA: Addison-Wesley, 2002.