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

Abstract

Let D be an infinite division ring, n a natural number and N a subnormal subgroup of GL n (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 maximal subgroup of GL n (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 GL n (D) cannot be polycyclic-by-finite.

On Almost Subnormal Subgroups and Maximal Subgroups in Skew Linear Groups

“…In the case when n = 1, we investigate maximal subgroups of an almost subnormal subgroup of D * . In [13], maximal subgroups of subnormal subgroups of D * was studied and it was shown that every nilpotent maximal subgroup of a subnormal subgroup of 36 TRUONG HUU DUNG D * is abelian [13,Theorem 2.3]. We extend this result for any maximal subgroup M of a non-central almost subnormal subgroup of D * in the case when D is infinite dimensional over its infinite center F and C D (M ) \ F contains an algebraic element over F .…”

“…Let D be a division ring with center F . Recently, some skew linear groups satisfying an identity was investigated [4,10,11,12,13]. For example, in [4] it was shown that every subnormal subgroup N of GL n (D) satisfying a generalized group identity over GL n (D) is central, i.e.…”

A note on solvable maximal subgroups in subnormal subgroups of ${\mathrm GL}_n(D)$

“…Also, the authors in [11] showed that if D is an infinite division ring, then D * contains no polycyclic-by-finite maximal subgroups. In the following corollary, we will see that every subnormal subgroup of D * does not contain non-abelian polycyclic-by-finite maximal subgroups.…”