Find all Sylow 3-subgroups of $S_3\times S_3$

Question

Find all Sylow 3-subgroups of $S_3\times S_3$?

This is what I already found: Since $o(S_3\times S_3)=36=2^2 3^2$ Sylow- $3$ subgroups have order $9$. If $n_3$ is the no. of Sylow- $3$ subgroups, Then $n_3|4$ and $3|(n_3 - 1)$. Hence $n_3$ should be $1$ or $4$. Now how can I find at least one subgroup of order $9$?

Answer 1

Hint $A_3 \times A_3 = (S_3 \times S_3)'$ is a normal subgroup of order $9$.

Answer 2

There's a normal one (as pointed out by Nicky Hekster).

Once that happens we know that there's only $1$, because they're all conjugate, by Sylow.

It's (isomorphic to) $\Bbb Z_3^2.$

Answer 3

The elements of order $3$ in $S_3\times S_3$ are: \begin{alignat}{1} &((123),()) \\ &((132),()) \\ &((),(123)) \\ &((),(132)) \\ &((123),(123)) \\ &((123),(132)) \\ &((132),(123)) \\ &((132),(132)) \\ \end{alignat} Jointly with $()$, they form a closed subset of $S_3\times S_3$, which is then a subgroup of order $9$, say $P_9$, clearly isomorphic to $C_3^2$. As there isn't any other element of order $3$, $P_9$ is unique.