The Semiprimitivity Problem for Twisted Group Algebras of Locally Finite Groups

Abstract

Let K[G] be the group algebra of a locally finite group G over a field K of characteristic p > 0. In this paper, we show that K[G] is semiprimitive if and only if G has no locally subnormal subgroup of order divisible by p. Thus we settle the semiprimitivity problem for such group algebras by verifying a conjecture which dates back to the mid 1970s. Of course, if G has a locally subnormal subgroup of order divisible by p, then it is easy to see that the Jacobson radical jK[G] is not zero. Thus, the real content of this problem is the converse statement. Our approach here builds upon a recent paper where we came tantalizingly close to a complete solution by showing that if G has no non-identity locally subnormal subgroup, then K[G] is semiprimitive. In addition, we use a two step process, suggested by certain earlier work on semiprimitivity, to complete the proof. The first step is to assume that all locally subnormal subgroups are central. Since this is easily seen to reduce to a twisted group algebra problem, our goal for this part is to show that Kt[G] is semiprimitive when G has no non-trivial locally subnormal subgroup. In other words, we duplicate the work of the previous paper, but in the context of twisted group algebras. As it turns out, almost all of the techniques of that paper carry over directly to this new situation. Indeed, there are only two serious technical problems to overcome. The second step in the process requires that we deal with certain extensions by finitary groups, and here we use recent results on primitive, finitary linear groups to show that the factor groups in question have well-behaved subnormal series. With this, we can apply previous machinery to handle the extension problem and thereby complete the proof of the main theorem.