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

Abstract

AbstractWe consider groups G such that the set $$[G,\varphi ]=\{g^{-1}g^{\varphi }|g\in G\}$$ [ G , φ ] = { g - 1 g φ | g ∈ G } is a subgroup for every automorphism $$\varphi $$ φ of G, and we prove that there exists such a group G that is finite and nilpotent of class n for every $$n\in \mathbb N$$ n ∈ 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.