group algebra

A group tells you how to multiply its elements, but you cannot add them. The group algebra fixes that: it lets you form formal sums of group elements with coefficients from a chosen ring, so 2g - 3h becomes a legal object. Multiplication extends the group's own operation by distributing across these sums. In one stroke the bare combinatorics of a group becomes a full-fledged ring you can do linear algebra inside.

Precisely, given a ring R and a group G, the group algebra R[G] is the free R-module with basis the elements of G, equipped with the multiplication determined by the group law: (sum a_g * g) times (sum b_h * h) equals sum over g, h of a_g * b_h * (g*h). It is commutative exactly when G is abelian and R is commutative. When R = k is a field this is a k-algebra of dimension |G| when G is finite.

The whole point is that R-linear representations of G are exactly the same data as R[G]-modules: a homomorphism G -> GL(V) extends uniquely to a ring homomorphism R[G] -> End(V). So representation theory becomes module theory over a single ring, and the structure of R[G] dictates how representations decompose. Maschke's theorem says k[G] is semisimple precisely when char(k) does not divide |G|.

Beyond finite groups the construction stays useful but subtler. For an infinite group R[G] uses finitely-supported sums; questions like whether C[G] has zero divisors (the Kaplansky conjectures) are deep and partly open. Variants such as the completed or topological group algebras drive harmonic analysis and the representation theory of compact and Lie groups.