For $H$ is a normal subgroup of $A_5$.
I know that the order of |aH| must divide both |$A_5$/H| and |a|. (Not entirely sure why it divides |a|, but that might be something to explain). From this we have to conclude that if |a| is relatively prime to all divisors of |A_5/H| then $a\in H$.
Is this just obvious once I explained why |aH| divides those things?
Also I have finally shown what the normal subgroups of $A_5$. But what sorts of homomorphic images of $A_5$ are possible?