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I'm trying to understand why how you can determine whether two groups of the form $\mathbb{Z}_n$ are isomorphic to each other. More specifically why is $\mathbb{Z}_2 \oplus \mathbb{Z}_3 \cong \mathbb{Z}_6$ but $\mathbb{Z}_3 \oplus \mathbb{Z}_5 \ncong \mathbb{Z}_{15}$?

Is there some method that I'm missing of determining that two groups are isomorphic?

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    (i) What you say you are trying to understand and your "more specifically" are not equivalent. (ii) $\mathbb{Z}_3\oplus\mathbb{Z}_5$ **is** isomorphic to $\mathbb{Z}_{15}$; what do you mean when you claim it is not?2012-04-25
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    Your examples are instances of the Chinese remainder theorem. http://en.wikipedia.org/wiki/Chinese_remainder_theorem2012-04-25
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    The title does not reflect the question.2012-04-25
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    Two canonical forms are provided by [the fundamental theorem of finitely generated abelian groups](http://en.wikipedia.org/wiki/Finitely-generated_abelian_group).2012-04-25

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