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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Jacobson radical of group rings of
locally finite groups

Author: D. S. Passman
Journal: Trans. Amer. Math. Soc. 349 (1997), 4693-4751
MSC (1991): Primary 16S34; Secondary 16S35, 20F50, 20F24
MathSciNet review: 1401781
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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.

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D. S. Passman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Keywords: Group algebras, twisted group algebras, Jacobson radical, semiprimitivity, locally finite groups
Received by editor(s): May 23, 1996
Additional Notes: Research supported by NSF Grant DMS-9224662.
Article copyright: © Copyright 1997 American Mathematical Society