Does the alternating group $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? What are all the possible orders of elements in the group $A_5$? Does $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? How about $\Bbb Z_{10}$? How about $\Bbb Z_5$? Justify your answers.

I've found that the possible orders are 1, 2, 3, and 5. I'm having a hard time conceptualizing this. 
The identity in the subgroup must map to the identity in $\Bbb Z_{20}$. There must exist a bijective function from the subgroup to $\Bbb Z_{20}$, and this function must satisfy $f(ab)=f(a)f(b)$. 
So I must find a function that works for this? I'm confused. 

Another hint:  Any element of $S_n$ can be written as a composition of disjoint cycles.  Further, the order of that composition will be the least common multiple of the length of each cycle within that composition.
Note that a $k$-cycle is an element of $A_5 \iff k$ is odd and $k \leq 5$.

Also note that Cayley's theorem guarantees that $\mathbb{Z}_5$ is a subgroup of $S_5$.  Would it also be a subgroup of $A_5$?
