Nilpotence and local nilpotence of linear groups

Abstract

Let GL( n, F) denote the general linear group over a commutative field F. It is well known that locally solvable subgroups of GL( n,F) are always solvable, but in general locally nilpotent subgroups need not always be nilpotent. The object of the present paper is to clarify this situation. For each odd prime p, let F p be a splitting field for X p − 1 over F; and let F 2 be a splitting field for X 4 − 1 over F. Put d p =[ F p : F], and let e p ⩽ ∞ denote the supremum of the set of integers e such that F p contains a primitive p e th root of 1. Then it is shown that: For n ⩾ 2 the locally nilpotent subgroups of GL( n,F) are all nilpotent if and only if e 2 < ∞ and e p < ∞ whenever pd p ⩽ n. Moreover, in the latter case there is a uniform bound on the nilpotence class, namely, max{ n( e p + 1)/ d p + 1| p = 2 or pd p ⩽ n}.