Minimal generating sets and Cayley graphs of Sylow \(p\)-subgroups of finite symmetric groups

Abstract

Minimal generating sets of a Sylow \(p\)-subgroup \(P_n\) of the symmetric group \(S_{p^n}\) are characterized. The number of ordered minimal generating sets of \(P_n\) is calculated. The notion of the type of a generating set of \(P_n\) is introduced and it is proved that \(P_n\) contains minimal generating sets of all possible type. The isomorphism problem of Cayley graphs of \(P_n\) with respect to their minimal generating sets is discussed.