Definition:Group Algebra

Definition
Let $\struct {k, + ,\circ}$ be a field.

Let $\struct {G, *}$ be a finite group.

Then the group algebra $k G$ or $k \sqbrk G$ is the set of all formal sums:


 * $\ds \sum_{g \mathop \in G} \alpha_g g : \alpha_g \in k$

That is, $k \sqbrk G$ is the free vector space over $k$ with basis $G$.