The degree of a class of minimal non-PS-groups (Q6655863)

Let \(G\) be a finite group. A subgroup \(H \leq G\) is \(\mathsf{s}\)-permutable (in \(G\)) if \(HS=SH\) for every Sylow subgroup \(S\) of \(G\). A group \(G\) is said to be an \(\mathsf{PS}\)-group if every cyclic subgroup of prime order or of order \(4\) is \(\mathsf{s}\)-permutable in \(G\); group \(G\) is said to be a minimal non-\(\mathsf{PS}\)-group if \(G\) is not a \(\mathsf{PS}\)-group but all its proper subgroups are \(\mathsf{PS}\)-groups.\N\NLet \(q\) be an odd prime. In the paper under review the author shows that the least \(n\) such that the minimal non-\(\mathsf{PS}\)-group \(G=\langle a,b \mid a^{q}=1, b^{4}=1, a^{b}=a^{-1} \rangle\) can be embedded in the symmetric group \(S_{n}\) is \(n=q+4\).