# Conjugacy class hypergroup

## Definition

Let $G$ be a FC-group (i.e., every conjugacy class is finite) and $R$ a ring. The conjugacy class hypoergroup or convolution algebra on conjugacy classes for $G$ over the ring $R$ is defined as follows:

• As a module, it is a $R$-module freely generated by the conjugacy classes in $G$.
• The structure constants for the multiplication are as follows: the coefficient of conjugacy class $k$ in the product of conjugacy classes $i$ and $j$ is the number of solutions to $ab = c$ where $c$ is a fixed element of $k$ and $a,b$ are arbitrary elements of $i,j$ respectively. (Note that we fix $c$ as a single element, and try to find all $a,b$ that work).

The convolution algebra on conjugacy classes is a subalgebra of the group algebra comprising those elements of the group algebra where the coefficients of conjugate elements of the group are equal.

Note that if $G$ is not a FC-group, a given element may be expressible as a product of elements of two conjugacy classes in infinitely many ways (the problem of coefficients being infinite) and a product of two conjugacy classes may involve infinitely many conjugacy classes (the problem of infinitely many nonzero coefficients in the expression for an element).

We shall denote the convolution algebra as $C(G,R)$.

## Image under a representation

The image of the convolution algebra on conjugacy classes under any irreducible linear representation over an algebraically closed field gives an algebra of scalar matrices. The matrices are scalar due to Schur's lemma. This yields that the scalar entries of these matrices are algebraic integers. Further information: Size-degree-weighted characters are algebraic integers

This fact is critical to the proof that the degree of any irreducible representation divides the order of the group.