Let $G=\langle g \rangle$ be a cyclic group of order $30$. Find the smallest positive integer $n$ such that there exists a monomorphism $\phi:G \rightarrow S_n$.
Attempt. Let $n$ be such an integer. Then $G\cong G/ker\phi\cong Im\phi$ by the first isomorphism thm, so $30=|Im\phi|$ divides $n!$ and $n\geq 5$. For $n=11$, the function $$\phi:G \rightarrow S_{11}:\phi(g^n)=(1~2~3~4~5)^n(6~7~8~9~10~11)^n$$ is a monomorphism. I don't know though if there are any other monomorphisms for $n=6,\ldots,10.$
Thanks in advance for the help.
Your problem is equivalent to finding an element of $S_n$ of order $30$. The cycle structure of such elements should include at least one of the following $30-$cycle, $15,2$ - cycle $10,3$ - cycle, $6,5$-cycle, or $3,2,5-$cycle, as $30$ should be the least common multiple of the length of the cycles. From this it's not hard to conclude that we need at least $10$ elements to be permuted, so the smallest number $n$ is $10$ and indeed
$$\phi(g) = (1,2,3)(4,5)(6,7,8,9,10) $$
induces such a monomorphism.