Given $S_{10}$, determine whether there is an cycle $\alpha \in S_{10}$ for which $\alpha^k =(1\ 2)(3\ 4)(5\ 6)(7\ 8)(9\ 10)$ for some $k\in \Bbb N$.
I am thinking of using the closure of $S_{10}$ as a group and that $\alpha^k$ is in itself an element of the group and therefore there should exist $\alpha$ for which it is true?
But I definitely do not know the correctness of my idea. Any help will be very appreciated.