Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups

TL;DR

This paper proves that nilpotent maximal subgroups of certain subnormal subgroups in skew linear groups are abelian and that such groups cannot have polycyclic-by-finite maximal subgroups, extending previous algebraic results.

Contribution

It generalizes prior results by showing nilpotent maximal subgroups are abelian and rules out polycyclic-by-finite maximal subgroups in skew linear groups.

Findings

Nilpotent maximal subgroups are abelian.

Maximal subgroups cannot be polycyclic-by-finite.

Extends previous algebraic group theory results.

Abstract

Let D be an infinite division ring, n a natural number and N a subnormal subgroup of GLn(D) such that n = 1 or the center of D contains at least five elements. This paper contains two main results. In the first one we prove that each nilpotent maximal subgroup of N is abelian; this generalizes the result in [R. Ebrahimian, J. Algebra 280 (2004) 244 - 248] (which asserts that each max- imal subgroup of GLn(D) is abelian) and a result in [M. Ramezan-Nassab, D. Kiani, J. Algebra 376 (2013) 1 - 9]. In the second one we show that a maximal subgroup of GLn(D) cannot be polycyclic-by-finite.