Smallest $n$ such that $S_n$ contains a subgroup isomorphic to $C_5$? The Cayley's theorem tells us that every finite group is isomorphic to a subgroup of some $S_n$
I have no idea how to go on about this question. I can list every elements of $S_1 , S_2, S_3, ... $ but I'm guessing that's not how it wants me to do it. I also have to find it for $C_2 \times C_2 \times C_2$ and $S_3 \times S_3$ but could anyone please explain the concept using the $C_5$ example? And maybe I will be able to try the rest myself.
Thank you.

Let be $a\in S_n$, $a^5=e$, $a\ne e$. By the theorem of Lagrange, $5\,\big|\,n!$ and we conclude that $n\ge 5$.
