Large totally symmetric sets (Q6179155)

A subset \(X\) of a group \(G\) is totally symmetric if any permutation of \(X\) can be realized by conjugation in \(G\). The first main results of the paper under review is Theorem 1.1: Let \(G\) be a group, and \(X \subset G\) a totally symmetric set of cardinality \(k>3\). Then \(|G| \geq (k+1)!\). If \(|G|=(k+1)!\), then \(G \simeq S_{k}\). The totally symmetric set \(X_{n}=\{(1,i) \mid i=2,\dots,n \} \subset S_{n}\) shows that the bound in Theorem 1.1 is sharp. The second main result shows that \(X_{n}\) is the only such example (with three exceptions for small \(n\)). Theorem 1.2: Let \(Y=\{y_{1},y_{2}, \dots, y_{k} \}\) be a totally symmetric set in \(S_{n}\) of cardinality \(k\). \begin{itemize} \item[(1)] If \(n \not \in \{3,4,6\}\) and \(k=n-1\), then \(Y\) is conjugate to \(X_{n}\). \item[(2)] If \(n=6\) and \(k=5\), then \(Y\) is conjugate to either \(X_{6}\) or \(\rho(X_{6})\) where \(\rho \in \mathrm{Out}(G)\) is non-trivial. \item[(3)] If \(n=4\) and \(k=3\), then \(Y\) is conjugate to \(X_{4}\) or \(\{(1,2),(1,3),(2,3)\}\), or \(Y\) is equal to \(\{(1,2)(3,4),(1,3)(2,4), (1,4)(2,3)\}\). \item[(4)] If \(n=3\), then \(Y\) may be any subset of any conjugacy class of \(S_{3}\). In particular \(k \leq 3\), and equality is realized by \(\{(1,2),(1,3),(2,3)\}\). \end{itemize}