The Jacobson radical of group rings of locally finite groups

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 ${\mathcal {J}}K[{G}]$ in terms of the radicals ${\mathcal {J}}K[{A}]$ of the group algebras of the locally subnormal subgroups $A$ of $G$. More precisely, we show that if $\operatorname {char} K=p>0$ and if $\mathbb {O}_{p}(G)=1$, then the controller of ${\mathcal {J}}K[{G}]$ is the characteristic subgroup $\mathbb {S}^{p}(G)$ generated by the locally subnormal subgroups $A$ of $G$ with $A=\mathbb {O}^{p’}(A)$. 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.