Maximal subgroups of GL n (D) with finite conjugacy classes

Abstract

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D * is an FC-group, then D is a field. Let M be a maximal subgroup of GL n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.