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Group algebras

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 ...

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