Group ring
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In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G.
R[G] can be described as the free module over R (if R is a field, this is just a vector space) generated by the elements g of G. The multiplication is the group operation in G extended by linearity and distributivity to the whole space.
The same notation is used for the elements of the group G and the basis elements of R[G], so that if in G we have g1g2 = g3, then the same holds true in R[G]. The whole structure of R[G] as an associative algebra over R follows when we apply the distributive law and R-linearity. The identity element of G serves as the 1 in R[G].
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An Example
Let G = Z3, the cyclic group of 3 elements with generator a. Then an element of C[G] is
- z1 + z2a + z3a2,
where z1, z2 and z3 are in <math>\mathbb{C}</math>, the complex numbers. If we take another element
- w1 + w2a + w3a2,
then their sum is
- (z1+w1) + (z2+w2)a + (z3+w3)a2
and their multiplication is
- (z1 + z2a + z3a2) (w1 + w2a + w3a2)
- = (z1w1 + z2w3 + z3w2) + (z1w2 + z2w1 + z3w3)a + (z1w3 + z2w2 + z3w1)a2.
In an example where G is a non-commutative group, we have to be careful to make the multiplication of the terms in the right order.
An example of a group ring of an infinite group is the ring of Laurent polynomials: this is exactly the group ring of the infinite cyclic group Z.
Representations
A module M over R[G] is then the same as a linear representation of G over the field R. There is no particular reason to limit R to be a field here. However, the classical results were obtained first when R is the complex number field and G is a finite group, so this case deserves close attention. It was shown that R[G] is a semisimple ring, under those conditions, with profound implications for the representations of finite groups. More generally, whenever the characteristic of R does not divide the order of the finite group G, then R[G] is semisimple (Maschke's theorem).
When G is a finite abelian group, the group ring is commutative, and its structure easy to express in terms of roots of unity. When R is a field of characteristic p, and the prime number p divides the order of the finite group G, then 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.
Category Theory
There is an elegant characterisation from category theory of the group ring construction as the left adjoint to the functor taking an associative R-algebra with one to its group of units.
Generalisation
Group algebras are more general algebras which derive their multiplication from the multiplication in G.ja:群環