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?

  • $\begingroup$ 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. $\endgroup$
    – quid
    Mar 12, 2015 at 0:07

1 Answer 1


Note that every cyclic group is also Abelian.

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).

  • $\begingroup$ Makes sense. thanks for telling me that every cyclic group is also abelian $\endgroup$
    – Justin
    Mar 12, 2015 at 6:00

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