THE JACOBSON RADICAL OF GROUP-RINGS OF LOCALLY FINITE-GROUPS

This paper is the final installment in a series of articles, started i n 1974, which study the semiprimitivity problem for group algebras K[G ] of locally finite groups. Here we achieve our goal of describing the Jacobson radical JK[G] in terms of the radicals JK[A] of the group al gebras of the locally subnormal subgroups A of G. More precisely, we s how that if char K = p > 0 and if O-p(G) = 1, then the controller of J K[G] is the characteristic subgroup S-p(G) generated by the locally su bnormal subgroups A of G with A = O-p' (A). In particular, we verify a conjecture proposed some twenty years ago and, in so doing, we essent ially solve one half of the group ring semiprimitivity problem for arb itrary groups. The remaining half is the more difficult case of finite ly generated groups. This article is effectively divided into two part s. The first part, namely the material in Sections 2-6, covers the gro up theoretic aspects of the proof and may be of independent interest. The second part, namely the work in Sections 7-12, contains the group ring and ring theoretic arguments and proves the main result. As usual , it is necessary for us to work in the more general context of twiste d group algebras and crossed products. Furthermore, the proof ultimate ly depends upon results which use the Classification of the Finite Sim ple Groups.