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Let $G_1$ and $G_2$ be finite groups, and let $n$ be the number of epimorphisms (i.e. surjective hom.) from $G_1$ to $G_2$.

Then what is the number of epimorphisms from $G_1$ to $G_2$ with distinct kernels? How?

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    @William: try counting the number of epimorphisms with the same kernel.2011-03-23

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The group $\operatorname{Aut}(G_2)$ acts on the set $E(G_1,G_2)$ of epis $G_1\to G_2$. You are trying to count the orbits. The action is without fixed points, so there are $|E(G_1,G_2)|/|\operatorname{Aut}(G_2)|$. But I don't think you can expect to express this in terms of $n$ only!

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    In fact one can drop the finiteness assumption in the sense that if one knows that $n$ (the number of epimorphisms) is finite then $Aut(G_2)$ is finite as well and the final answer remains unchanged.2011-03-23