Definition:Group Algebra

Definition

Let $\left({k, + ,\circ}\right)$ be a field.

Let $\left({G, *}\right)$ be a finite group.

Then the group algebra $k G$ or $k \left[{G}\right]$ is the set of all formal sums:

$\displaystyle \sum_{g \in G} \alpha_g g\ :\ \alpha_g \in k$

That is, $k \left[{G}\right]$ is the free vector space over $k$ with basis $G$.