A question says: prove or prove or disprove that there are only 2 non-isomorphic abelian groups of order 2009.
I think that it is true because... I split up $2009$ into $7 \times 7 \times 41$ and so this gives the groups $\mathbb{Z}_{2009}$ $\mathbb{Z}_{7} \oplus \mathbb{Z}_{7\times 41}$ where the first order divides the next by the fundamental theorem of finite abelian groups (and we can prove these are not isomorphic by using order of elements) but then I thought about the group $\mathbb{Z}_{7}\oplus \mathbb{Z}_{7} \oplus \mathbb{Z}_{41}$. What actually IS this group? By FTFAG 7 doesn't divide 41 so surely it can't be another decomposition. And 7,7 is not coprime so it can't be isomorphic to $\mathbb{Z}_{2009}$. But maybe I'm wrong? Argh!