The General Linear Group $GL(n,\mathbb{F}_p)$ has an interesting property that the proof of Sylow theorem for this group can be given which is based on the natural action of this group on the $n$-dimensional vector space over $\mathbb{F}_p$. A short sketch of proof is:
1) The group of upper uni-triangular matrices over $\mathbb{F}_p$ is a Sylow-p subgroup.
2) $GL(n,\mathbb{F}_p)$ acts on the collection $\mathcal{F}=\{0\subseteq V_1\subseteq V_2\cdots \subseteq V_n\}$ of chains, where $V_i$ is $i$-dimensional vector space over $\mathbb{F}_p$, and the Stabilizers of chains which induce identity map on quotient $V_i/V_{i-1}$ are precisely the Sylow-$p$ subgroups; Sylow-$p$ subgroups are conjugate.
3) The number of Sylow-$p$ subgroups is equal to the number of chains in above collection(my guess); and it can be shown easily by combinatorics that $|\mathcal{F}|\equiv 1(p)$.
The details are given in the book of Alperin-Bell, Groups and Representations.
Question: Instead of $GL(n,\mathbb{F}_p)$, consider the family of symmetric (permutation) groups $S_n$. Can we prove Sylow theorems for this group in some similar way as done for $GL(n,\mathbb{F}_p)$?
The existence of Sylow-p subgroup in $S_n$ is already done by Cauchy, which is expressed by "wreath product", and a proof is given in Herstein-Topics in Algebra. But I couldn't see (or prove) their "conjugacy" and "cardinality is $1$ mod $p$"
[Here, I want to see a proof without following the arguments of any well known proof of Sylow theorems for arbitrary finite groups.]