For what values n is the group $S_n$ cyclic.

So, a friend of mine and I were confused about when $S_n$ is cyclic.

I thought $S_n$ is cyclic for $S_{2n}$ where $n$ is an integer and abelian for $n \leq 2$ but my friend said its the other way around.

Can anyone confirm?

• What do you mean by $S_n$? If it is the permutation group on $n$ symbols you are both wrong. See the answer.// and you are wrong regardless the group since it is impossible for a group to be cyclic while not abelian.
– quid
Mar 12, 2015 at 0:07

For $n \geq 3$, $(12)$ and $(23)$ do not commute so $S_n$ is non-Abelian (and thus is not cyclic).
For $n < 3$, $S_n$ is cyclic (and hence Abelian).