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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?

iou
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    See this answer by Ted for an IMO intresting description of Sylow $p$-subgroups of $S_{p^n}$ as kind of graph automorphisms. The case $p=2$ when we get these groups as automorphisms of the binary tree is particularly nice. Anyway, you get this group as the $n$-fold wreath product $(C_p\wr(\cdots(C_p\wr C_p)\cdots)\wr C_p)\wr C_p$. – Jyrki Lahtonen Jan 25 '17 at 21:20
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    You can also show this inductively, by dividing ${1,\ldots,p^n}$ into $p$ subsets $\Omega_1,\ldots,\Omega_p$ of length $p^{n-1}$, have $p$ copies of a Sylow $p$-subgroup $P$ of $S_{p^{n-1}}$ act on each $\Omega_i$ and then $C_p$ acts on the set ${\Omega_1,\ldots,\Omega_p}$ to show that the Sylow $p$-subgroup of $S_{p^n}$ is $P\wr C_p$. – Robert Chamberlain Jan 25 '17 at 21:28

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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.

Asinomás
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