TITLE: Determining all Sylow $p$-subgroups of $S_n$ up to isomorphism?
QUESTION [8 upvotes]: I'm trying to understand a classification of all Sylow $p$ subgroups of $S_n$. 
Let $Z_p$ be the subgroup of $S_p$ generated by $(12\cdots p)$. Then $Z_p\wr Z_p$ has order $p^p\cdot p=p^{p+1}$, and is isomorphic to a subgroup of $S_{p^2}$. 
Define inductively $Z_p^{\wr r}$ by $Z_p^{\wr 1}=Z_p$ and $Z_p^{\wr k+1}=Z_p^{\wr k}\wr Z_p$. It is easy to show by induction that $Z_p^{\wr r}$ has order $p^{(p^{r-1}+p^{r-2}+\cdots+1)}$, and since by inductively assuming $Z_p^{\wr r-1}$ is isomorphic to a subgroup of $S_{p^{r-1}}$ and $Z_p$ isomorphic to a subgroup of $S_p$, then $Z_p^{\wr r}$ is isomorphic to a subgroup of $S_{p^r}$. 
However, I can't make the jump that if $n=a_0+a_1p+\cdots+a_kp^k$ is the base $p$ expansion, then any Sylow $p$-subgroup is isomorphic to 
$$
\underbrace{Z_p^{\wr 1}\times\cdots\times Z_p^{\wr 1}}_{a_1}\times
\underbrace{Z_p^{\wr 2}\times\cdots\times Z_p^{\wr 2}}_{a_2}\times\cdots\times
\underbrace{Z_p^{\wr k}\times\cdots\times Z_p^{\wr k}}_{a_k}.
$$
I know this group has order
$$
(p)^{a_1}(p^{p+1})^{a_2}\cdots(p^{(p^{k-1}+p^{k-2}+\cdots+1)})^{a_k}=p^{\sum_{i=1}^k a_i(1+\cdots+p^{i-1})}=p^{\nu_p(n!)}
$$
which is the order of any Sylow $p$-subgroup of $S_n$, based on the formula here. However, I couldn't find an epimorphism from any Sylow $p$-subgroup onto this product, or vice versa. Is it clear how this is isomorphic to a subgroup of $S_n$? Then I understand that the isomorphism would just follow from the Sylow theorems. Thanks.

REPLY [3 votes]: Given two permutation groups H on n points and K on m points, there is a permutation group called H × K that acts on n + m points.  The group is abstractly the direct product of the two groups, and the action is very simple: an element like $(h,k)$ acts on the first n points exactly like h did on its n points, and on the last m points exactly like k did on its m points.
For instance the Sylow 3-subgroup of Sym(6) is $\langle (1,2,3) \rangle \times \langle (4,5,6) \rangle$.