Group bijections commuting with inner automorphisms (Q6631331)

``As showed in [\textit{A. A. Simonov} et al., Sib. Math. J. 65, No. 3, 627--638 (2024; Zbl 07918234); translation from Sib. Mat. Zh. 65, No. 3, 577--590 (2024)], when in the generalized Alexander quandle with the operation \[x\circ y =\phi(xy^{-1})y,\ x, y\in G\] constructed from some group \(G\) by using \(\phi\in Aut G\), we replace \(\phi\) with an antiautomorphism, the construction enables us to construct a quandle if and only if\N\[\Nx\phi(y)x^{-1} = \phi(xyx^{-1})\tag{1}\N\]\Nfor all \(x, y\in G\). The following problem is posed in [\textit{A. A. Simonov} et al., Sib. Math. J. 65, No. 3, 627--638 (2024; Zbl 07918234); translation from Sib. Mat. Zh. 65, No. 3, 577--590 (2024)]:\N\N\textbf{Problem 1}. Given a group \(G\), describe all bijections \(\phi\) of \(G\) onto itself satisfying (1) for all \(x, y\in G\). Moreover, some full description of such bijections is given for the symmetric group \(S_3\) on three letters and for absolutely free groups in [\textit{A. A. Simonov} et al., Sib. Math. J. 65, No. 3, 627--638 (2024; Zbl 07918234); translation from Sib. Mat. Zh. 65, No. 3, 577--590 (2024)]. Condition (1) also appeared in [\textit{P. M. Cohn} Free rings and their relations. 2nd ed. London etc.: Academic Press (1985; Zbl 0659.16001)], where Cohn considered an equivalent definition of a skew field constructed by an unary operation \(\phi\) on a multiplicative group. The construction requires that three extra conditions apart from (1) be satisfied. The function \(\phi\) itself is neither an automorphism nor an antiautomorphism, but some special bijection satisfying three extra conditions. In this article we consider the set of bijections \(\phi\in B(G)\) satisfying (1) on an arbitrary group \(G\) and establish some general properties of \(B(G)\) for every \(G\). In particular, we show that the composition of mappings makes \(B(G)\) a group. As an application, we obtain a full description of \(B(G)\) for all dihedral groups. Closing the article, we consider some possible generalizations of Problem 1 for quandles and groups.''\N\NFor details about quandles and Alexander quandles see [\textit{V. G. Bardakov} et al., Monatsh. Math. 184, No. 4, 519--530 (2017; Zbl 1466.57004)] and [\textit{M. Elhamdadi} et al., J. Algebra Appl. 11, No. 1, Article ID 1250008, 9 p. (2012; Zbl 1248.20070)].