A note on engel groups and local nilpotence

Abstract

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class script C sign including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class script C sign is both finite-of-exponent-e(n)-by-nilpotent-of-class≤ c(n) and nilpotent-of-class≤ c(n)-by-finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.