Central Series of Nilpotent Subgroups of the Unitary Groups

Abstract

Let Gm(R) be a nilpotent subgroup of the general unitary group of m x m matrices over the ring R where R is a principal ideal domain with involution and non-2 characteristic. The purpose of this paper is to calculate the upper central series of Gm(R) and determine whether or not it coincides with its lower central series. This is done by determining the structure of products and commutators of matrices in the group and applying induction on the c’th centre of Gm(R). In the case that R is the Gaussian integers, we then consider prime factorisations of its entries to find conditions for when the upper and lower central series do not coincide.