Throughout is a finite group. For the definition of the group algebra where is a field, see this post.

**Remark 1**. If is a representation of then is a -module. Conversely, if is a -module, then defined by is a representation of

*Proof*. Let and We define

The above definition is extended to every element and by

Proving that defines a -module structure (also called a -module) over is easy: we only need to show that for any and we have

and

Both of the above are trivially true: the first one holds because and thus is linear. The second one holds because is a group homomorphism and thus

For the converse, we need to show that for all and that is a group homomorphism. It is clear that is linear. Also, if then and hence So Finally, is a group homomorphism because if and then

**Definition**. A representation is called **irreducible** if with the structure defined in Remark 1, is a simple -module. If is not irreducible, it is called **reducible**.

**Remark 2**. Every degree one representation of is irreducible.

*Proof*. Let where be a degree one representation of Then, since a -submodule of is obviously a -submodule of and has no non-trivial -submodule. So is a simple -module and hence is irreducible.