Maximal subgroups of \(\text{GL}_n(D)\) with finite conjugacy classes.
Let \(D\) be an infinite division ring and \(M\) a maximal subgroup of the matrix group \(\text{GL}_n(D)\), for some positive integer \(n\). Suppose that \(M\) is an FC-group, i.e. each \(g\in M\) has a finite number of conjugates in \(M\). The paper under review shows that then \(M\) is the multiplicative group of some maximal subfield \(M'\) of the matrix ring \(M_n(D)\); in addition, it proves that if \(M'\) is an algebraic extension of the centre \(Z(D)\) of \(D\), then \(D\) is a finite-dimensional algebra over \(Z(D)\). The crucial step towards the authors' proof of this result is to show that the group \(M\) is Abelian.
scientific article; zbMATH DE number 5627042
scientific article