Question: > 
> Does $S\_{10}$ have a subgroup that is isomorphic to $\Bbb{Z}/30\Bbb{Z}$?
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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: 
**Hint:** What would the cycle type of a generator of such a subgroup be?
The question is equivalent to

> 
> Does $S\_{10}$ have an element of order $30$ ?
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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 $30=\operatorname{lcm}(2,3,5)$.