Sylow subgroup of $S_{p^2}$ Find Sylow p-subgroup (at least 1) of $S_{p^2}$.
We know that $|S_{p^2}|=(p^2)!$ and hence $|P|=p^{p+1}$ (where P is p-subgroup).
How can I describe this subgroup?

Consider the subgroup of permutations on $1,2,3,\dots, p^n$ that permute the subsets $\{1,2,3,\dots,p\},\{p+1,p+2,\dots,2p\}, \{2p+1,2p+2,\dots, 3p\},\dots, \{p(p-1)+1,\dots,p^2\}$ cyclically, and  also permute the elements inside each group cyclically.
There are $p^p$ ways to permute the subsetes internally and there are $p$ ways to permute them externally. So this subgroup has $p^{p+1}$ elements.
