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?

Question

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:

Answer 1

Sylow subgroups of $G\times H$ are also products of Sylow subgroups of $G$ and of $H$. Now what are the Sylow subgroups of $S_3$?

Answer 2

Quang's method is undoubtedly the easier approach here, although there are alternative ways of attacking this problem.

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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?

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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?

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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.