THE SEMIPRIMITIVITY PROBLEM FOR TWISTED GROUP-ALGEBRAS OF LOCALLY FINITE-GROUPS

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 semipr imitive if and only if G has no locally subnormal subgroup of order di visible by p. Thus we settle the semiprimitivity problem for such grou p 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 ze ro. Thus, the real content of this problem is the converse statement. Our approach here builds upon a recent paper where we came tantalizing ly close to a complete solution by showing that if G has no non-identi ty locally subnormal subgroup, then K[G] is semiprimitive. In addition , we use a two step process, suggested by certain earlier work on semi primitivity, to complete the proof. The first step is to assume that a ll locally subnormal subgroups are central. Since this is easily seen to reduce to a twisted group algebra problem, our goal for this part i s to show that K-t[G] is semiprimitive when G has no non-trivial local ly subnormal subgroup. In other words, we duplicate the work of the pr evious paper, but in the context of twisted group algebras. As it turn s out, almost all of the techniques of that paper carry over directly to this new situation. Indeed, there are only two serious technical pr oblems to overcome. The second step in the process requires that we de al with certain extensions by finitary groups, and here we use recent results on primitive, finitary linear groups to show that the factor g roups in question have well-behaved subnormal series. With this, we ca n apply previous machinery to handle the extension problem and thereby complete the proof of the main theorem.