I'm having some trouble determining whether or not two groups are isomorphic to each other, and disproving using some structural property. The problems I've been having trouble with are:
For the following groups determine whether or not they are isomorphic. If they are, give an isomorphism. If not, disprove by giving some structural property that distinguishes them.
a) $\mathbb{Z}_8$ x $\mathbb{Z}_2$ and $\mathbb{Z}_4$ x $\mathbb{Z}_4$
b) $\mathbb{Z}_3$ x $\mathbb{Z}_5$ and $\mathbb{Z}_{15}$
The product of the numbers are the same, so I'm assuming they are of the same cardinality and a bijection exists. But beyond looking at cardinalities/bijections, I'm lost as to how to actually "see" an isomophism and if there's an distinguishing structural property. Any help would be greatly appreciated, as I would really like to understand this topic very concretely.