## Definition For a $k$-algebra $A$ and an (abelian) group $G$, the space of group morphisms with finite support $$A[G]:=hom_{fin\,supp}(G,A)$$ (i.e. space containing just those morphisms which send only finitely many elements of $G$ not to $0$) equipped with the multiplication defined by convolution of functions is called *group algebra of $A$ over $G$*. With pointwise addition $K[G]$ is an associative $A$-algebra. Equivalently one can think of the multiplication inthe following way Take the elements of a group, $G$, as labelling a module basis and define the multiplication of two elements by $x_g \cdot x_h:=x_{gh}$. If $A$ is a ring one calls this construction also a *group ring*. The notion of group algebra is a special case of that of a [[category algebra]]. Any group algebra is in particular a [[Hopf alebra]] and a [[graded algebra]]. ### The convolution product 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, $$\Big(\sum_{g\in G}n_ge_g\Big)\Big(\sum_{g\in G}m_ge_g\Big) = \sum_{g\in G}\Big(\sum_{g_1\in G}n_{g_1}m_{g_1^{-1}g}e_g\Big).$$ ## Properties +-- {: .num_remark} ###### Remark There is an [[adjunction]] $$(R[-]\dashv (-)^\times):Alg_R \stackrel{(-)^\times}{\to}Grp$$ where $R[-]$ forms group rings and $(-)^\times$ assigns to an $R$-algebra its [[group of units]]. =-- +-- {: .num_remark} ###### Remark Let $V$ be an abelian group. A morphism of rings $K[G]\to End(V)$ of the group ring to the endomorphism ring of $V$ is a $K[G]$-module. And any morphism of groups $p:G\to End(V)$ can by extended to a morphism of rings $P:K[G]\to End (V)$ by $p(g)\mapsto P(e_g)$. This observation is used extensively in the theory of [[group representation|group representations]]. =-- +-- {: .num_remark} ###### Remark (Maschke's theorem) Let $G$ be a finite group, let $K$ be a field. Then $K[G]$ is a [[semi-simple algebra]] iff the order of $G$ is not divisible by the [[characteristic]] of K. =-- [[!redirects group ring]] [[!redirects group algebras]]