Number of homomorphism Recently, I was reading on group operations and came across a question like determine the number of group homomorphism for two groups. I don't specifically recall the example but how can one determine the number of homomorphism? Wouldn't that depend on the order of the group? and bigger the order, more would be the no. of homomorphism. The example that I have in mind is rather simple, determine the number of homomorphism from $\mathbb{Z_{8}}\rightarrow \mathbb{Z_{10}}$?
Figuring out the number of homomorphisms from a cyclic group is relatively easy; once you know what happens to the generator, you immediately know what everything else in the group is mapped to.  All that remains from there is to check which of those give you a homomorphism.

In this case, the question we need to ask is if $f:\mathbb{Z}_8 \to \mathbb{Z}_{10}$, what values of $f(1)$ could give us a homomorphism?

Keeping in mind that we need to have $8[f(1)]\equiv 0\pmod{10}$ (why?), the only valid choices are $f(1) \equiv 5$ and $f(1) \equiv 0$. You can verify that both of these define a valid homomorphism.  Thus, there are two homomorphisms in this instance.
order of ${Z_8}=8$,  order of ${Z_{10}}=10$. Both are finite cyclic groups. Hence, to check number of Homomorphism between them is gcd of their orders, $\text{gcd}(8,10) = 2$, Hence the number of homomorphism function is $2$. This is not coincident.

One of The property of homomorphism states ${\phi{(g^n)= (\phi(g))^n}}$, such a homomorphism is completely specified by the image of 1. That is, if, 1 maps to a then x maps to xa.
Another property states that if ${|g|}$ is finite, then ${|\phi(g)|}$ divides ${|g|}$, so ${|a|} ={1,2}$ as then ${a={0,5}}$, i.e., ${1 \mapsto 0}$ or ${1 \mapsto 5}$ .These are the only two homomorphism is possible.