are of finite support, and the ring axioms are readily verified.
are sometimes[1] written as what are called "formal linear combinations of elements of
Let G = C3, the cyclic group of order 3, with generator
, their sum is and their product is Notice that the identity element 1G of G induces a canonical embedding of the coefficient ring (in this case C) into C[G]; however strictly speaking the multiplicative identity element of C[G] is 1⋅1G where the first 1 comes from C and the second from G. The additive identity element is zero.
Consider the group ring RQ, where R is the set of real numbers.
An arbitrary element of this group ring is of the form where
For example, Note that RQ is not the same as the skew field of quaternions over R. This is because the skew field of quaternions satisfies additional relations in the ring, such as
To be more specific, the group ring RQ has dimension 8 as a real vector space, while the skew field of quaternions has dimension 4 as a real vector space.
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, and its group of invertible elements contains a subgroup isomorphic to G. For considering the indicator function of {1G}, which is the vector f defined by the set of all scalar multiples of f is a subring of R[G] isomorphic to R. And if we map each element s of G to the indicator function of {s}, which is the vector f defined by the resulting mapping is an injective group homomorphism (with respect to multiplication, not addition, in R[G]).
If G is a finite group of order greater than 1, then R[G] always has zero divisors.
is prime, then G has no nonidentity finite normal subgroup (in particular, G must be infinite).
As a set and vector space, it is the free vector space on G over the field K. That is, for x in K[G], The algebra structure on the vector space is defined using the multiplication in the group: where on the left, g and h indicate elements of the group algebra, while the multiplication on the right is the group operation (denoted by juxtaposition).
The group algebra, consisting of finite sums, corresponds to functions on the group that vanish for cofinitely many points; topologically (using the discrete topology), these correspond to functions with compact support.
, or The dimension of the vector space K[G] is just equal to the number of elements in the group.
The field K is commonly taken to be the complex numbers C or the reals R, so that one discusses the group algebras C[G] or R[G].
This result, Maschke's theorem, allows us to understand C[G] as a finite product of matrix rings with entries in C. Indeed, if we list the complex irreducible representations of G as Vk for k = 1, .
Assembling these mappings gives an algebra isomorphism where dk is the dimension of Vk.
The subalgebra of C[G] corresponding to End(Vk) is the two-sided ideal generated by the idempotent where
These form a complete system of orthogonal idempotents, so that
is closely related to Fourier transform on finite groups.
For a more general field K, whenever the characteristic of K does not divide the order of the group G, then K[G] is semisimple.
When K is a field of characteristic p which divides the order of G, the group ring is not semisimple: it has a non-zero Jacobson radical, and this gives the corresponding subject of modular representation theory its own, deeper character.
The center of the group algebra is the set of elements that commute with all elements of the group algebra: The center is equal to the set of class functions, that is the set of elements that are constant on each conjugacy class If K = C, the set of irreducible characters of G forms an orthonormal basis of Z(K[G]) with respect to the inner product Much less is known in the case where G is countably infinite, or uncountable, and this is an area of active research.
[3] The case where R is the field of complex numbers is probably the one best studied.
In this case, Irving Kaplansky proved that if a and b are elements of C[G] with ab = 1, then ba = 1.
Whether this is true if R is a field of positive characteristic remains unknown.
This conjecture is equivalent to K[G] having no non-trivial nilpotents under the same hypotheses for K and G. In fact, the condition that K is a field can be relaxed to any ring that can be embedded into an integral domain.
The conjecture remains open in full generality, however some special cases of torsion-free groups have been shown to satisfy the zero divisor conjecture.
In general, group rings contain nontrivial units.
The above adjunction expresses a universal property of group rings.