$Z_5$ isomorphic to a subgroup of $S_4$ The question asks me if $Z_5$ is isomorphic to a subgroup of $S_4$. 
What I was thinking of doing is writing down all the elements of $S_4$ and then again finding the subgroups generated by every element. But that is just very long. I assume there should be a proper way to check this. Any hint?

No. First, we know $|S_4|=4!=24,$ and $|\Bbb{Z}_5|=5$. Then it is clear that $5$ does not divide $24,$ and by Lagrange's theorem, the order of any subgroup of $S_4$ must divide $24.$ So no such subgroup can exist.
