The group algebra of a finite group G is a vector space of dimension |G| which also carries extra structure involving the product operation on G. In a sense, group algebras are the source of all you need to know about representation theory. In particular, the ultimate goal of representation theory – that of understanding all the representations of finite groups – would be achieved if group algebras could be fully analysed. Group algebras are therefore of great interest.
After defining the group algebra of G, we shall use it to construct an important faithful representation, known as the regular representation of G, which will be explored in greater detail later on.
The group algebra of G
Let G be a finite group whose elements ...