We take a break from representation theory to discuss some topics in group theory which will be relevant in our further study of representations. After defining conjugacy classes, we develop enough theory to determine the conjugacy classes of dihedral, symmetric and alternating groups. At the end of the chapter we prove a result linking the conjugacy classes of a group to the structure of its group algebra.
Throughout the chapter, G is a finite group.
Let x, y ∈ G. We say that x is conjugate to y in G if
The set of all elements conjugate to x in G is
and is called the conjugacy class of ...