Math Help - Homomorphisms

1. Homomorphisms

Find ALL group homomorphisms from f: Z/2Z --> Z/4Z.

I know elements of Z/2Z have order 1 or 2.
I know elements of Z/4Z have order 1, 2, or 4.

I believe I must use this to find all the homomorphism. I don't know what to do from here. Any advice? Thanks

2. Re: Homomorphisms

Originally Posted by jzellt
Find ALL group homomorphisms from f: Z/2Z --> Z/4Z.

I know elements of Z/2Z have order 1 or 2.
I know elements of Z/4Z have order 1, 2, or 4.

I believe I must use this to find all the homomorphism. I don't know what to do from here. Any advice? Thanks
The maps are determined by where $1$ gets mapped to. Try to prove that $1\mapsto x$ implies $|x|\mid 2$. Moreover, prove that if $|x|\mid 2$ then $1\mapsto x$ gives a well-defined morphism.

In geneneral, it's helpful to remember that as abelian groups $\text{Hom}(\mathbb{Z}_m,\mathbb{Z}_n)\cong \mathbb{Z}_{(m,n)}$.

3. Re: Homomorphisms

to amplify what mr. drexel is saying:

we know f(0) = 0. so the only choice we have is where f sends 1.

4 choices:

f(1) = 0
f(1) = 1
f(1) = 2
f(1) = 3.

what goes wrong when we send 1-->1 (think: what does f(1+1) have to be)?