On ECP-Groups

TL;DR

This paper investigates finite groups where every subgroup is conjugate-permutable, providing answers to key questions, including examples and structural properties of such groups.

Contribution

It answers open questions about ECP-groups, shows that all exponent 3 groups are ECP, and explores their structural classifications.

Findings

Every group of exponent 3 is an ECP-group

Existence of non-regular ECP-3-groups

The class of finite ECP-groups is neither a formation nor a variety

Abstract

According to T. Foguel a subgroup $H$ of a group $G$ is called conjugate-permutable if $ HH^x=H^xH$ for every $x\in G$. Mingyao Xu and Qinhai Zhang studied finite groups with every subgroup conjugate-permutable (ECP-groups) and asked three questions about them. We gave the answers on these questions. In particular, every group of exponent 3 is ECP-group, there exist non-regular ECP-$3$-groups and the class of all finite ECP-groups is neither formation nor variety.