On finite groups in which the twisted conjugacy classes of the unit element are subgroups

TL;DR

This paper investigates groups where the twisted conjugacy classes of the identity form subgroups, revealing the existence of finite nilpotent groups of arbitrary class and an infinite nonnilpotent example, thus disproving a conjecture.

Contribution

It constructs examples of groups with the property that twisted conjugacy classes are subgroups, including finite nilpotent groups of any class and an infinite nonnilpotent group, challenging existing conjectures.

Findings

Existence of finite nilpotent groups of arbitrary class with the property

Existence of an infinite nonnilpotent group with the property

Disproof of conjecture 18.14 in [5]

Abstract

We consider groups $G$ such that the set $[G,\varphi]=\{g^{-1}g^{\varphi}|g\in G\}$ is a subgroup for every automorphism $\varphi$ of $G$, and we prove that there exists such a group $G$ that is finite and nilpotent of class $n$ for every $n\in\mathbb N$. Then there exists an infinite nonnilpotent group with the above property and the conjecture 18.14 of $[5]$ is false.