Maximal subgroups of \(\text{GL}_n(D)\)
In this interesting paper the authors continue their study of maximal subgroups of skew linear groups. They start by exhibiting a metabelian maximal subgroup of the multiplicative group of the quaternions. Suppose \(M\) is a maximal subgroup of the multiplicative group \(D^*\) of the division ring \(D\) with centre \(F\). If \(M\) is nilpotent they show that \(M\) is metabelian, if \(M\) is nilpotent and \(M\setminus F\) contains an element algebraic over \(F\) they prove that \(M\) is Abelian and if \(M\) is non-Abelian, locally soluble and algebraic over \(F\), they describe \(M\) in some detail. Then they focus their attention on the maximal subgroups \(M\) of the general linear group \(\text{GL}(n,D)\) for \(n\) a positive integer. If \(M\) is locally finite, they show that \(D=F\) is a locally finite field. If \(D\) is infinite and locally finite-dimensional over \(F\) and if \(M\) is locally nilpotent, they prove that \(M\) is Abelian. We refer the reader to the paper where there are further interesting results.
scientific article; zbMATH DE number 1891319
scientific article