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.
One crucial way to distinguish groups from one another is looking at the orders of elements. If one group has more elements of a certain order than another group, they cannot be isomorphic.
There is no general easy way to prove groups are isomorphic. Perhaps you know the Chinese Remainder Theorem?