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.

Received: 20 September 2024 | Accepted: 23 April 2025