The Jacobson radical of group rings of locally finite groups
D. S. Passman
Trans. Amer. Math. Soc. 349 (1997), 4693-4751.
Abstract. This paper is the final installment in a series of articles, started
in 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 algebras of the
locally subnormal subgroups A of G. More precisely, we show that if
char K = p > 0 and if O_p(G) = 1, then the controller of JK[G] is the
characteristic subgroup S^p(G) generated by the locally subnormal 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 essentially solve one half of the group
ring semiprimitivity problem for arbitrary groups. The remaining half is the
more difficult case of finitely generated groups. This article is effectively
divided into two parts. The first part, namely the material in Sections 2-6,
covers the group 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 twisted group
algebras and crossed products. Furthermore, the proof ultimately depends upon
results which use the Classification of the Finite Simple Groups.