# On groups where the twisted conjugacy class of the unit element is a subgroup

## TL;DR

This paper investigates groups where the conjugacy class of the identity element under any automorphism forms a subgroup, revealing structural properties like nilpotency and bounds on nilpotency class.

## Contribution

It proves that such groups with finitely generated conditions have bounded verbal width in their lower central series and are nilpotent if they satisfy the descending chain condition.

## Key findings

- Lower central series members have finite verbal width bounded by generators and series index.
- Groups satisfying the descending chain condition are proven to be nilpotent.
- For finite abelian-by-cyclic groups, an upper bound on the nilpotency class is established.

## Abstract

We study groups $G$ where the $\varphi$-conjugacy class $[e]_{\varphi}=\{g^{-1}\varphi(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $\varphi$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th member of the lower central series has a finite verbal width bounded in terms of $n,k$. Moreover, we prove that if such group $G$ satisfies the descending chain condition for normal subgroups, then $G$ is nilpotent. Finally, if $G$ is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of $G$.