Subgroups of $S_n$

Question

I am pretty confused right now with this. There should be a subgroup of $S_n$ with order x. So for example, for which n does $S_n$ contain a subgroup of order 60? What about if we are looking for a cyclic subgroup of order 60?

Answer 1

In general it is not easy to determine the values of $x$ in which $S_n$ has a subgroup of order $x$. For the cyclic case it is equivalent to find $x$ such that $S_n$ has an element of order $x$, which is quite straightforward:

if $m_i$ for $1\le i\le k$ are distinct positive integers and $\sum{m_i} \le n$ then for any subgroup $I$ of $\{1,...,k\}$ the group $S_n$ has an element (cyclic subgroup) of order $x = lcm (m_j )_{j \in I}$. For example in $S_9$, we have elements of orders $1,2,3,4,5,6,7 ,8,9, 10,12,14,15,20$.