Let $${\displaystyle G}$$ be a group, written multiplicatively, and let $${\displaystyle R}$$ be a ring. The group ring of $${\displaystyle G}$$ over $${\displaystyle R}$$, which we will denote by See more

1. Let G = C3, the cyclic group of order 3, with generator $${\displaystyle a}$$ and identity element 1G. An element r of C[G] can be written as See more

Group algebras occur naturally in the theory of group representations of finite groups. The group algebra K[G] over a field K is essentially … See more

Adjoint
Categorically, the group ring construction is left adjoint to "group of units"; the following functors are an adjoint pair: See more

Using 1 to denote the multiplicative identity of the ring R, and denoting the group unit by 1G, the ring R[G] contains a subring isomorphic to R, … See more

Much less is known in the case where G is countably infinite, or uncountable, and this is an area of active research. The case where R is the field … See more

If a group has a length function – for example, if there is a choice of generators and one takes the word metric, as in Coxeter groups – … See more