On locally finite p-groups satisfying an Engel condition

Abstract

For a given positive integer n and a given prime number p, let r = r (n,p) be the integer satisfying pr-1 < n ≤ pr. We show that every locally finite p-group, satisfying the n-Engel identity, is (nilpotent of n-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either pr or pr-1 if p is odd. When p = 2 the best upper bound is pr-1 pr or pr+1. In the second part of the paper we focus our attention on 4-Engel groups. With the aid of the results of the first part we show that every 4-Engel 3-group is soluble and the derived length is bounded by some constant.