Is there any cyclic subgroup of order 6 in in $ S_6$? Is there any cyclic subgroup of order 6 in   $ S_6$?
Attempt:
$|S_6|=6!=720$
Let $H$ be a subgroup of $S_6$ ,$H$ cyclic $\iff\langle H \rangle=\{e,h,h^2,...,h^{n-1}\}=S_6$

Yes: no need of great theorems. The subgroup generated by $(1,2,3,4,5,6)$ does the job.
