Semiprimitivity of group algebras of locally finite groups. II (Q1919567)

The result of this paper is an intermediate contribution to the author's project to obtain a criterion (in group terms) of semiprimitivity of the group algebra of a locally finite group [for part I cf. the author and the reviewer, Proc. Lond. Math. Soc., III. Ser. 67, No. 2, 243-276 (1993; Zbl 0798.16015)]. In this paper the author considers the locally finite groups that have a finite subnormal series with factors which are either infinite simple, or generated by locally subnormal subgroups or contain no element of order \(p\). Let \(K\) be a field of characteristic \(p\). The main theorem says that the group algebra \(K[G]\) is semiprimitive (i.e. the Jacobson radical of \(K[G]\) is 0) if and only if \(G\) contains no locally subnormal subgroup of order a multiple of \(p\). (A finite subgroup \(X\) of \(G\) is called locally subnormal if \(X\) is subnormal in every finite subgroup containing \(X\)).