On the semisimplicity of modular group algebras

Abstract

Let G be a discrete group, let K be a field and let KG denote the group algebra of G over K. We say that KG is semisimple if its Jacobson radical JKG is zero. If K has characteristic 0 and K is not an algebraic extension of the rationals then it is known [l, Theorem 1 ] that KG is semisimple. Moreover it appears likely that in the remaining characteristic 0 cases we also have semisimplicity. Thus nothing particularly interesting occurs here. If K has characteristic p>0 and G is a p'-group (that is, G has no elements of order p), then it is known (see [2]) that KG has no nil ideals and that for suitably big fields the group algebra KG is semisimple. Again it appears that for the remaining fields we also have semisimplicity. The interest in characteristic p stems from the fact that, unlike the case in which G is finite, it is quite possible for G to have elements of order p and yet have the group algebra KG be semisimple. Several examples of such groups have been exhibited and in each case a big abelian group is involved as either a subgroup or a factor group. In this paper we study the group algebras KG of those groups G having a big abelian subgroup or factor group. The methods used are extremely elementary. Two interesting examples of the type of groups which we can deal with are as follows. Let P be a cyclic group of order p and let C be an infinite cyclic group. Let Gi = C P and G2=P I C where I denotes the restricted Wreath product. Now Gi has a normal torsion free abelian subgroup of finite index p and G2 has a normal elementary abelian p-subgroup E with G2/E infinite cyclic. Surprising as it may seem if K is an algebraically closed field of characteristic p, then 7<Gi and KG2 are both semisimple. For the remainder of this paper K is an algebraically closed field of characteristic p. By a linear 7I-character of KG we mean a FJ-homomorphism X: KG—*K.