Finding the number of abelian groups of order 10

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.

• Your understanding is correct, there is only one. – user491874 Nov 20 '17 at 17:56