How many group homomorphisms are there from $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z}$ to $\mathbb{Z}/18\mathbb{Z} $
I know that a generator should be mapped to a generator in a homomorphism. But I am unable to proceed in this problem.
Any help is appreciated. Thanks in advance.
Any such will be determined by where it sends $(1,0,0),(0,1,0)$ and $(0,0,1)$. And the orders of the images divide $3,4$ and $9$ respectively. There are $3$ elements of order dividing $3$, $2$ of order dividing $4$, and $9$ of order dividing $9$, by cyclicity. So, $3\cdot2\cdot9=54$.