Just a thought from the logical side of things. You have:
So we can start by assuming that G is NOT isomorphic to Z10. This is a common approach to proofs of the form P -> (Q v R), where "v" is the logical operator "or".
In other words, not being Z10 means no elements have order ten. But the order of an element must divide the order of the group, so look for elements of orders 5 and 2.