On multiplicative subgroups in division rings. (Q2797005)

Let \(D\) be a division ring with centre \(F\) and multiplicative group \(D^*\). Considerable information, gathered by many authors, has been amassed about the maximal subgroups and the subnormal subgroups of \(D^*\). Here the authors consider the maximal subgroups of the subnormal subgroups of \(D^*\). Their substantial results are often very technical and hence are not suitable for a brief summary. We confine ourselves to a few titbits with a view to giving at least the flavour of this work.NEWLINENEWLINE Suppose \(M\) is a non-Abelian maximal subgroup of the subnormal subgroup \(G\) of \(D^*\). Assume that either a) \(M\) is metabelian or b) \(D\) is locally finite-dimensional over \(F\) and \(M\) contains no non-cyclic free subgroups. Then \(D=F(M)\) (even \(=F[M]\) in case a)) and there is a maximal subfield \(K\) of \(D\) such that \(K\) is Galois over \(F\) with \(M/(K^*\cap G)\cong\text{Gal}(K/F)\) a finite simple group (hence of prime order in case a)) and with \(K^*\cap G\) the Fitting subgroup of \(M\).NEWLINENEWLINE The authors' final section contains further results on a maximal subgroup \(M\) of \(D^*\), especially when \(M\) is also locally soluble or locally nilpotent.