Maximal subgroups of \(\text{GL}_1(D)\)
Let \(D\) be finite dimensional division algebra. By a well-known result of I.~N.~Herstein, every finite normal subgroup of the multiplicative group \(D^*\) of \(D\) is central. The authors extend this by proving that every finitely generated normal subgroup of \(D^*\) is central. The authors then study maximal subgroups of \(D^*\) for \(D\) a not necessarily finite dimensional division algebra. They give a number of results, some involving interaction (or rather a lack of it) between the degree and the characteristic of \(D\) and some rather technical. As a snippet, to give something of the flavour of this second part of the paper, we quote the following. Let \(D\) be a division ring that is algebraic over its centre. If \(D^*\) has a soluble maximal subgroup, then \(D\) is locally finite-dimensional. If \(D^*\) has an Abelian maximal subgroup, then \(D\) is finite dimensional.
scientific article; zbMATH DE number 1340554
scientific article