Tuple regularity and \(\mathrm{k} \)-ultrahomogeneity for finite groups (Q6634500)

Let \(G\) be a finite group and \(k\in \mathbb{N}\). The group \(G\) is \(k\)-ultrahomogeneous (\(k\)-tuple regular) if for all tuples \((x_{1}, \ldots, x_{k}), \, (y_{1},\ldots, y_{k}) \in G^{k}\) for which the assignment \(x_{1} \mapsto y_{1}, \ldots, x_{k} \mapsto y_{k}\) defines an isomorphism \(\varphi\) between \(X=\big \langle x_{1},\ldots, x_{k} \big \rangle\) and \(Y=\big \langle y_{1}, \ldots, y_{k} \big \rangle\), there exists an automorphism \(\widehat{\varphi} \in \mathrm{Aut}(G)\) with \(\widehat{\varphi}|_{X}=\varphi\) (there exists a bijection \(\Psi \!: G \rightarrow G\) such that, for every \(x \in G\), the extended assignment \(x_{1} \mapsto y_{1}, \ldots x_{k} \mapsto y_{k}, g\mapsto \Psi(x)\) defines an isomorphism between \(\big \langle x_{1},\ldots, x_{k}, x \big \rangle\) and \(\big \langle y_{1}, \ldots y_{k}, \Psi(x) \big \rangle\)). The group \(G\) is ultrahomogeneous if it is \(k\)-ultrahomogeneous for every \(k\in \mathbb{N}\).\N\NThe main results of the paper under review are the following two classification theorems:\N\NTheorem A. Let \(G\) be a finite group.\N\begin{enumerate}\N\item If \(G\) is solvable, then \(G\) is \(2\)-tuple regular if and only if \(G=A\times B\), where \(A\) and \(B\) have coprime orders, \(A\) is an abelian group with homocyclic Sylow subgroups and one of the following holds for \(B\): \begin{itemize}\item \(B\) is isomorphic to one of the groups in \(\{1, Q_{8}, G_{64}, A_{4}; C_{32}\rtimes Q_{8}; \mathrm{SL}(2,3), G_{192}\}\) where \(G_{64}\) denotes the Suzuki \(2\)-group of order \(64\) and \(G_{192}\) is a certain group of order 192; \item \(B \simeq M \times C_{2^{n}}\), where \(M\) is an abelian group of odd order with homocyclic Sylow subgroups and the cyclic group acts on \(M\) by inversion. \end{itemize}\N\N\item If \(G\) is non-solvable, then \(G\) is \(2\)-tuple regular if and only if \(G=H \times E\), where \(H\) and \(E\) have coprime orders, \(H\) is an abelian group with homocyclic Sylow subgroups and \(E \in \big \{ \mathrm{SL}(2,5), \mathrm{PSL}(2,5), \mathrm{PSL}(2,7) \big \}\). \end{enumerate}\N\NComparing the classification given in Theorem A to the classification of ultrahomogeneous finite groups in [\textit{G. Cherlin} and \textit{U. Felgner}, J. Lond. Math. Soc., II. Ser. 62, No. 3, 784--794 (2000; Zbl 1016.20013)], the author obtains the following equalities between the group classes.\N\NTheorem B. Let \(G\) be a finite group. Then the following are equivalent. \begin{enumerate} \item \(G\) is \(k\)-ultrahomogeneous for some \(k \geq 2\); \item \(G\) is \(\ell\)-tuple regular for some \(\ell \geq 2\); (iii) \(G\) is ultrahomogeneous. \end{enumerate}\N\NIn contrast to this, the author remarks that there exist \(1\)-ultrahomogeneous finite groups that are not ultrahomogeneous (see [\textit{J. Zhang}, J. Algebra 153, No. 1, 22--36 (1992; Zbl 0767.20009)]).