On finite groups in which the twisted conjugacy classes of the unit element are subgroups (Q6614077)

Let \(G\) be a group and \(\varphi\) be an automorphism of \(G\). The \(\varphi\)-conjugacy class of an element \(g \in G\) is the set \[[g]_{\varphi}=\{x^{-1}gx^{\varphi} \mid x \in G\}.\] In particular, if \(e\) is the unit element of \(G\), then \[[e]_{\varphi}=[G,\varphi]=\{x^{-1}x^{\varphi} \mid x \in G \}.\]\N\NThe groups in which \([e]_{\varphi}\) is a subgroup for each \(\varphi \in \mathrm{Aut}(G)\) have been studied, for example in [\textit{D. L. Gonçalves} and \textit{T. Nasybullov}, Commun. Algebra 47, No. 3, 930--944 (2019; Zbl 1441.20023)].\N\NThe main result in the paper under review is Theorem 1.2: For every integer \(n \in \mathbb{N}\) and for every odd prime \(p\), there exists a finite \(p\)-group \(G\) of class \(n\) in which the \(\varphi\)-conjugacy class of the unit element is a subgroup for every \(\varphi \in \Aut(G)\).\N\NThanks to Theorem 1.2, the author constructs an interesting example of a non-nilpotent group \(G\) such that \([e]_{\varphi}\) is a subgroup for every \(\varphi \in \Aut(G)\) (see Corollary 3.1). This disproves a conjecture of Bardakov, Neshchadim, Nasybullov in [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 18th edition. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. 227 p. (2014; Zbl 1372.20001), Problem 18.14].