A note on CII groups and CCII groups

Abstract

A group G is CII or, equivalently, 2-Engel if [g, h]= [g^-1;h^-1] for all elements g and h in G, and is CCII if the central quotient G/Z(G) is CII. In this paper, we give sufficient conditions and necessary conditions for a group to be CCII. In particular, we show that every CCII group is nilpotent of class at most 4 and list all CII groups and all CCII groups of order n with n < 64 up to isomorphism.