I have a few questions related to applying properties of group homomorphisms.
1) How many homomorphisms are there from onto ?
Well, since both are cyclic, each has to map to a generator, so is it a trick question, I think there is only one, determined by
First, the above does NOT define a group homomorphism( for example, check what happens with
There's no epimorphism as required but you haven't yet proved it
2) How many homomorphisms are there from to ?
Well, gcd(8,20) = 4 so there are 4?
I get, for example, ...and I can't find any more.
What's missing here and which would help you also with (1)?? Hint: think of orders of elements under homomorphisms.
I am pretty sure this works for the groups of integers under addition modulo n...
I am not actually certain what properties the groups need to have such that you can count the number of homomorphisms by taking the gcd of the order of the groups.
Any help/confirmation appreciated! Thank you.