The Upper Central Series in Linear Groups

Abstract

In [15] M.S. Garaščuk proves that every locally nilpotent linear group is hypercentral. K.W. Gruenberg ([19]) gave another proof of this and, amongst other things, proved that the Fitting subgroup of a linear group is nilpotent and that the central height of a linear group is less than 2ω. In [20] he sharpened the latter result; there exists an integer-valued function ψ(n) such that if G is any subgroup of GL(n, F) then G has central height at most ω + ψ(n). This bound is reduced in [66], the correct bound being given for the locally nilpotent case. Chapter 8 consists of an exposition of the results of these papers together with some related theorems (on the size of the upper central factors of a linear group) and examples.