The Nilpotency Class of Finitely Generated Groups of Exponent Four

Abstract

Wright (Theorem 3 of [6]) has shown that for every positive integer n an n-generator group of exponent 4 is nilpotent of class at most 3n − 1. When n is 2 this result is best possible (see, for example, §3 of [4]). On the other hand, if for some positive integer m the nilpotency class of every m-generator group of exponent 4 were 3m − 3, then groups of exponent 4 would all be soluble [3] and there would be a positive integer k such that for every positive integer n every n-generator group of exponent 4 would have nilpotency class at most n + k (Theorem A of [2]). The gap between these two results is intriguingly small. In this paper we report on some further narrowing of this gap which makes the problem of trying to close the gap still more attractive. We prove the following result.