Given a group 
, the algebra 
is a vector space
 |
of finite sums of elements of 
,
with multiplication defined by 
,
the group operation. It is an example of a group ring.
For example, when the group is the symmetric group on three letters, 
,
the group ring 
is a six-dimensional algebra. An example of the product
of elements is
 |
Modules over 
correspond to complex group
representations of 
.
When 
is a finite
group then 
is a finite-dimensional algebra.
