On the Number of Generators and Composition Length of Finite Linear Groups

Abstract

In 1991 Dixon and Kovacs 8 showed that for each field K which has finite degree over its prime subfield there is a number d such that every K finite nilpotent irreducible linear group of degree n 2 over K can be ' generated by d n log n elements. Afterwards Bryant et al. 3 proved K that the same is true for solvable linear groups and this led to asking whether a similar result could hold also removing the solvability hypothesis. In 15 it is proved that the answer is positive in the particular case of finite fields. In the present paper we are able to deal with the case of number fields, thus giving a complete solution to the problem. Namely, Ž . denoting by d G the number of generators of a group G, we prove: