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

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 not nilpotent group with the above property and the Conjecture 18.14 of Khukhro and Mazurov (The Kourovka Notebook No. 20, 2022) is false.