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Find the number of abelian groups of order $10$.

So I tried using the abelian fundamental theorem.. You can write $10= 2 \times 5$ But $p(1)*p(1)=1$. So is there only one abelian group of order $10$? With $ \mathbb{Z} _{10} $ isomorphic to $ \mathbb{Z} _{2} \times \mathbb{Z} _{5}$? I think I have an understanding issue here.

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    $\begingroup$ Your understanding is correct, there is only one. $\endgroup$ – user491874 Nov 20 '17 at 17:56

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