Artinian modules over hyperfinite groups (Q1307041)

A group \(G\) is said to be hyperfinite if it has an ascending normal series with finite factors. The structure of Artinian modules over hyperfinite locally soluble groups is described in this article. In particular, the following result is proved. Let \(F\) be a field, \(G\) a hyperfinite locally soluble group and \(A\) a monolithic Artinian \(FG\)-module such that \(C_G(A)=\{1\}\). (1) If \(F\) has characteristic \(0\), then the socle \(S\) of \(G\) contains a subgroup \(J\) such that \(S/J\) is locally cyclic and the core of \(J\) in \(G\) is trivial. (2) If \(F\) has prime characteristic \(p\), then the subgroup \(P=O_p(G)\) is nilpotent with finite exponent and there exists a normal subgroup \(H\) of \(G\) containing \(P\) such that \(H/P\) is finite and the socle \(\overline S\) of \(\overline G=G/J\) is a \(p'\)-group containing a subgroup \(\overline J\) such that \(\overline S/\overline J\) is locally cyclic and \(\overline J\) has trivial core in \(\overline G\).