Which are the nine Sylow $2$-subgroups of $S_3 \times S_3$? What is the only Sylow $3$-subgroup of $S_3 \times S_3$? And the most important...Why? Which are the nine Sylow 2-subgroups of $S_3 \times S_3$? What is the only Sylow 3-subgroup of $S_3 \times S_3$? And the most important... why?
You can find excellent information about the Sylow theorems in the website of the scholars John Beachy and William Blair. Here is the link: 
 A: Quang's method is undoubtedly the easier approach here, although there are alternative ways of attacking this problem.

Sylow $2$-subgroups have order $4$ in $S_3 \times S_3$.  However, none of them can be isomorphic to $\mathbb{Z}_4$ because this would imply that there exists an element of order $4$ in $S_3 \times S_3$.  Can you see why such an element cannot exist?
Thus, every Sylow $2$-subgroup will be isomorphic to the Klein-four group, $\mathbb{Z}_2 \times \mathbb{Z}_2$.  Such a subgroup will be generated by two elements of order $2$ whose product is also of order $2$.  An example of such a subgroup is $H = \langle (e, 12);(13,e) \rangle$.  How many ways are there of constructing a similar subgroup?

As for Sylow $3$-subgroups of $S_3 \times S_3$, the Sylow theorems tell us that there are either $4$ of them, or only $1$ of them.  Furthermore, each Sylow $3$-subgroup will have order $9$.
Now, it is a fact that any group of order $p^2$ is isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_p \times \mathbb{Z}_p$.  However, I claim that it's not possible to have an element of order $9$ in $S_3 \times S_3$, so the latter must be the case.
What is special about $\mathbb{Z}_p \times \mathbb{Z}_p$?  Every non-identity element has order $3$.  How can we use this information to conclude that only a single Sylow $3$-subgroup exists?

Useful facts for considering the above:

*

*In any direct product of groups, the order of an element $(g_1, g_2) \in G_1 \times G_2$ is the least common multiple of the orders of $g_1$ and $g_2$ in their respective groups.


*Any permutation in $S_n$ can be written as a product of disjoint cycles, and the order of that permutation will be the least common multiple of the cycle lengths.
A: Sylow subgroups of $G\times H$ are also products of Sylow subgroups of $G$ and of $H$. Now what are the Sylow subgroups of $S_3$?
