Semiprimitivity of group algebras of locally finite groups. II

Abstract

Let K be a field of characteristic p > 0, let G be a locally finite group, and let K[ G] denote the group algebra of G over K. In this paper we study the Jacobson radical JK[ G] when G has a finite subnormal series with factors which are either p′-groups, infinite simple, or generated by locally subnormal subgroups. For example, we show that if such a group G has no finite locally subnormal subgroup of order divisible by p, then JK[ G] = 0. The argument here is a mixture of group ring and group theoretic techniques and requires that we deal more generally with twisted group algebras. Furthermore, the proof ultimately depends upon certain consequences of the classification of the finite simple groups. In particular, we use J.I. Hall's classification of the locally finite finitary simple groups.