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.

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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.