The composition and derived lengths of a soluble group

Abstract

Given a finite soluble group with derived length d and composition length n, the present paper investigates upper bounds for d in terms of n. An elementary argument is used to show that d⩽⌜ 2n 3 ⌝, where ⌜ 2n 3 ⌝ denotes the least integer greater than or equal to 2n 3 . The sharper bound d⩽⌜ (n + 3) 2 − 3 (n + 2) ⌝ is obtained by using properties of soluble subgroups of two-dimensional general linear groups. Finally, arguments like those used by Hall and Higman are used in conjunction with bounds for the derived length of soluble linear groups to show that d⩽ f( n) < 3log 2 n + 9.