Solution
= Solution
Since $|S_3|=6$, its Sylow $2$-subgroups have order two and its Sylow $3$-subgroups have order three. They are
$$
\boxed{\langle(12)\rangle,\quad\langle(13)\rangle,\quad\langle(23)\rangle}
$$
and
$$
\boxed{A_3=\langle(123)\rangle},
$$
respectively. The latter is the unique Sylow $3$-subgroup.