number of homomorphisms from a group to another

Question

how do I find the number of homomorphisms from one group G1 to another group G2 does it matter if G1 and G2 are not cyclic? for example there is a question that says: how many homomorphisms are there from Z20 to Z16? the answer is 4 but how do I get them?

Answer 1

For general groups $G_1$ and $G_2$, this can be a quite challenging problem.

In your particular case, however, you can reason as follows. A homomorphism $\phi$ from $\mathbb Z_{20}$ to $\mathbb Z_{16}$ is completely determined by specifying where the image of the generator $z$ of $\langle z \rangle \cong \mathbb Z_{20}$ maps to. The only condition is that $z^{20}$, which is trivial in $\mathbb Z_{20}$, must map into a trivial element. This amounts to $\phi(z^{20}) = \phi(z)^{20}$ being trivial. In the group $\mathbb Z_{16}$, this is the same as requiring $\phi(z)^4$ being trivial. Now, the latter condition means that $z$ must map into an element in $\mathbb Z_{16}$ whose order divides $4$. You can check that there are precisely $4$ such elements: $\{ 0, 4, 8, 12 \}$.