Is every subgroup of $S_n$ cyclic?

Question

I've just showed that every subgroup of $S_3$ is cyclic.

I'm not familiar with permutation groups yet and I'm curious to know whether every subgroup of $S_n$ is cyclic.

As a counterexample might be $S_4$ probably, but I'm not sure at all.

Thanks in advance for explanation.

Answer 1

The Klein subgroup of $S_4$ is a non cyclic proper subgroup of order $4$. All its elements have order $2$.

Answer 2

If you mean that all proper subgroups of $S_n$ are cyclic, it would imply that all proper subgroups are abelian. Now it is quite easy to show that $A_n$ is not abelian for $n\ge 4$.

Answer 3

No.

By Cayley's Theorem, every finite group is isomorphic to a subgroup of a symmetric group $S_n$ for some $n$ (dependent upon the finite group).

Answer 4

$S_4$ has subgroups of orders 1,2,3,4,6,8,12 and 24. Except subgroups of orders 1,2 and 3 (these groups are always cyclic) and some subgroups of order 4, all other subgroups are non cyclic.

Answer 5

That is a property that cyclic groups possess. And while there are examples, such as $S_3$, of non-cyclic groups which possess it also, $S_n, n\gt3$ are not among them.

As already remarked, for instance, $S_k\le S_n\,,\forall k\lt n$.