Minimal generating sets and Cayley graphs of Sylow $p$-subgroups of finite symmetric groups

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.