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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.

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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?

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