I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements.
Obviously a group with 2007 elements has a subgroup of order 9 (sylow theorem) and I think this should help me with finding the homomorphism, but I'm stuck. I'm also thinking about using the fact that there also should be a subgroup of order 223 which is cyclic. But this didn't help me much either.
$223$ is a prime number. By Sylow theorems, if $N$ is the number of $223$-sylow subgroups, then $$N \equiv 1 \mod (223) $$ $$N \mid 3^2$$ This implies that $N =1$.
Thus there is an unique $223$-sylow , which is therefore normal. Suppose we call it H.
Consider $G/H$, it is a group of order $9$, and the projection $$\phi : G \to G/H $$ is a surjective homomorphism. The action of the homomorphism is $$\phi(g) = gH$$ i.e t sends $g$ in its lateral class in $G/H$.