## Definition For a $k$-algebra $A$ and an (abelian) group $G$ the group ##Idea Take the elements of a group, $G$, as labelling a basis for a vector space over a field $k$, then that multiplication of the group will extend to give that vector space the structure of an [[algebra]] over $k$. This is usually denoted $k[G]$. If instead of a field we used the ring of integers, $\mathbb{Z}$, it is usual to call the result the __group ring__. For this we take the free abelian group on the set of elements of the group and extend the multiplication to give a ring structure on the result. This is usually denoted $\mathbb{Z}[G]$. As a further generalisation, the field can be replaced by any commutative ring. ## The multiplication: If we denote by $e_g$, the generator corresponding to $g\in G$, then an arbitrary element of $k[G]$ can be written as $\sum_{g\in G}n_ge_g$ where the $n_g$ are elements of $k$, and only finitely many of them are non-zero. The multiplication is then by what is sometimes called a 'convolution' product, that is, $$\Big(\sum_{g\in G}n_ge_g\Big)\Big(\sum_{g\in G}m_ge_g\Big) = \sum_{g\in G}\Big(\sum_{g_1\in G}n_{g_1}m_{g_1^{-1}g}e_g\Big).$$ ##Thoughts Thinking of a [[group]], $G$, as a special sort of [[category]], the **group algebra** (denoted $k[G]$ or $k G$) of a group is just the [[category algebra]] of that category. ##Extra structure The group algebra is always a [[Hopf algebra]]. The group algebra is always a [[graded algebra]]. Sometimes instead of working over a [[ground field]] $k$, one allows $k$ to be a [[commutative unital ring]]. Then we talk about __group ring__ (though it is in fact a commutative unital $k$-[[commutative unital algebra|algebra]]). The integer group ring $\mathbb{Z} G$ is the most important example, extensively used in the [[representation theory]] of finite groups. [[!redirects group ring]] [[!redirects group algebras]]