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. 

Hint: a subgroup isomorphic to ${\mathbb Z}_n$ would be generated by an element of order $n$.
