Twisted conjugacy classes of the unit element

TL;DR

This paper investigates the structure of groups where the twisted conjugacy class of the identity element forms a subgroup under all automorphisms, extending previous results from abelian groups to more general cases.

Contribution

It characterizes groups in which the twisted conjugacy class of the identity is a subgroup for every automorphism, broadening understanding beyond abelian groups.

Findings

Twisted conjugacy class of the identity is a subgroup in certain groups.

The structure of such groups is explicitly characterized.

Extension of Fel'shtyn and Troitsky's results to non-abelian groups.

Abstract

We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).