Determine the least positive integer $n$ for which the cyclic group $(\Bbb Z_{60},+\pmod{60})$ is isomorphic to a subgroup of $S_n$.

Question

Determine the least positive integer $n$ for which the cyclic group $(\Bbb Z_{60},+\pmod{60})$ is isomorphic to a subgroup of $S_n$.

I have absolutely no idea how to even start this problem. I looked at Cayley's theorem and I think the answer is $S_{60}$ because each element must be mapped to another one. The other possibility is a permutation group of $S_5$ of order $60$ but that doesn't make sense because not every element would be mapped to a unique element in the permutation set.

Please help

Answer 1

Hint. You need a symmetric group with an element of order $60$. Think about how to calculate the order of a permutation when you write it as a product of disjoint cycles.

Answer 2

Hint: The orders of $(1,2,3), (4, 5,6,7), (8,9,10,11,12)$ are $3$, $4$, and $5$, respectively. They are disjoint cycles.