I'm afraid to say that there is no `general rule'. I mean, for your example I would prove that every subgroup of is isomorphic to (can you see why this is sufficient?). But if I was to prove that , the dihedral group of order 4, was isomorphic to the Klein 4-group I would point out that it is not cyclic, and there are only two groups of order 4, one of which is cyclic, the other if the Klein 4-group.

However, when trying to find an isomorphism there are a few things to note,

If you map to the generators then your homomorphism will be surjective,

If the kernel is trivial then your homomorphism will be injective,

Try to find a `common' group which both groups are isomorphic to (as in the example),

Often there exists some well-known function which actually turns out to be an isomorphism of groups (for example, , and the isomorphism is given by ).

A homomorphism is an isomorphism if and only if there exists an inverse function. Sometimes this can be easily found.