Skew linear groups with rank conditions

Abstract

Let D be a division ring, n a positive integer, and GLn(D) the general skew linear group of degree n over D. In this paper, we study the structure of a subgroup G of GLn(D) satisfying some rank conditions. Among results, we prove that if G is locally nearly radical of finite co-central rank, then G has a normal solvable subgroup N of finite co-central rank such that G/N is finite. Also, we show that G is locally generalized radical of finite rank if and only if G contains a normal solvable subgroup S of finite rank such that G/S is finite. These facts generalize some our previous results.