Question: Give a representative of each Isomorphism class of Abelian group of order 225. Which ones are cyclic?
By the Fundamental theorem of finite abelian group:
$\left | G \right |=225=3^{2}5^2$
Now, the possible isomorphism are
$G\cong \mathbb{Z}_{3}\otimes \mathbb{Z}_{3}\otimes \mathbb{Z}_{5}\otimes \mathbb{Z}_{5}, \mathbb{Z}_{9}\otimes \mathbb{Z}_{25}, \mathbb{Z}_{9}\otimes \mathbb{Z}_{5}\otimes \mathbb{Z}_{5}, \mathbb{Z}_{3}\otimes \mathbb{Z}_{3}\otimes \mathbb{Z}_{25}$
Here's my confusion:
My solution indicates the ONLY cyclic groups to be $\mathbb{Z}_{9}\otimes \mathbb{Z}_{25}$, but from the theorem of external direct product of cyclic group, this cannot be the only cyclic group since
for all cyclic groups G and H, $G \otimes H$ are cyclic IFF $GCD\left ( \left | H \right |,\left | G \right | \right )$ are coprime.
Any help is appreciated.
There's only one cyclic group of any order. $9$ and $25$ are coprime. $\mathbb Z_9\oplus\mathbb Z_{25}$ is cyclic.
I can only guess that $\oplus$ or $\times$ is what you mean, but one of them must be because $\otimes$ is nonsensical in this case, as $\mathbb Z_9\otimes\mathbb Z_{25}=0$.