I take it that an epimorphism is a surjective homomorphism?

is the hint you are giving me, that the order of an image under the homomorphism must divide the order of the group being mapped into?

ie, if , then

divides

is that true? I think so!

So there are no surjective homomorphisms, because there would need to be an element of order 8 in to map to an element of order 8 in

to get the whole group? Is there some homomorphism property which dictates that?

Sure! if is a group hom. and , then
and there is 3 homomorphisms to

, because elements of

have order 1,2,4,5,10,20 and elements of

have orders 1,2,4, and 8, and those have 3 possible orders in common?!

Yeppers
Coincidentially?, Euler's phi function of 8 is 3....?

Not, it is

Tonio
Is any/some/all of this reasoning valid?

Thank you!!