Group Algebra

The group algebra K[G], where K is a field and G a group with the operation *, is the set of all linear combinations of finitely many elements of G with coefficients in K, hence of all elements of the form


where a_i in K and g_i in G for all i=1,...,n. This element can be denoted in general by

 sum_(g in G)a_gg,

where it is assumed that a_g=0 for all but finitely many elements of g.

K[G] is an algebra over K with respect to the addition defined by the rule

 sum_(g in G)a_gg+sum_(g in G)b_gg=sum_(g in G)(a_g+b_g)g,

the product by a scalar given by

 asum_(g in G)a_gg=sum_(g in G)(aa_g)g,

and the multiplication

 (sum_(g in G)a_gg)(sum_(g in G)b_gg)=sum_(g in G,h in G)(a_gb_h)g*h.

From this definition, it follows that the identity element of G is the unit of K[G], and that K[G] is commutative iff G is an Abelian group.

If the field K is replaced by a unit ring R, the addition and the multiplication defined above yield the group ring R[G].

If G=Z, and * is the usual addition of integers, the group ring R[G] is isomorphic to the ring R[x^(-1),x] formed by all sums


where n,m are integers, and a_i in R for all indices i=n,...,m.

Let G be a locally compact group and mu be a left invariant Haar measure on G. Then the Banach space L^1(G) under the product given by the convolution (f*g)(s)=int_Gf(t)g(t^(-1)s)dmu(t) for s in G is a commutative Banach algebra that is called the group algebra of G.

See also

Algebra, Group, Semigroup Algebra

Portions of this entry contributed by Margherita Barile

Portions of this entry contributed by Mohammad Sal Moslehian

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Bonsall, F. F. and Duncan, J. Complete Normed Algebras. New York: Springer-Verlag, 1973.

Referenced on Wolfram|Alpha

Group Algebra

Cite this as:

Barile, Margherita; Moslehian, Mohammad Sal; and Weisstein, Eric W. "Group Algebra." From MathWorld--A Wolfram Web Resource.

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