Definition:Group Algebra

Definition
Let $(k,+,\circ)$ be a field.

Let $(G,*)$ be a finite group.

Then the group algebra $kG$ or $k[G]$ is the set of all formal sums


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

That is, $k[G]$ is the free vector space over $k$ with basis $G$.