Does $S_{10}$ have a subgroup isomorphic to $\Bbb{Z}/30\Bbb{Z}$?

Question

Does $S_{10}$ have a subgroup that is isomorphic to $\Bbb{Z}/30\Bbb{Z}$?

I tried to use the fact that if such subgroup $H$ exists, then $|H|=|\Bbb{Z}/30\Bbb{Z}|=30$, however I don't see why such subgroup can't exist.

Beyond that I really have no idea how to proceed. Can anyone give a hint?

Answer 1

Hint: What would the cycle type of a generator of such a subgroup be?

Answer 2

The question is equivalent to

Does $S_{10}$ have an element of order $30$ ?

Now, $30=2 \cdot 3 \cdot 5$ and $2+3+5 \le 10$, and so the answer is yes: just take a permutation with cycle structure $2-3-5$. The simplest one is $$ (1,2)(3,4,5)(6,7,8,9,10) $$ This permutation has order $30=\operatorname{lcm}(2,3,5)$.