Group algebra

Summary

The group algebra $F[G]$ of a group $G$ over a field $F$ is the vector space with basis $G$. It is a ring, with the group multiplication extended linearly. Its modules are exactly the representations of $G$ over $F$. So representation theory of groups becomes module theory of a single ring. Over a commutative ring $R$ in place of $F$, the same construction gives the group ring $R[G]$.