Thread: 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

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 gets mapped to. Try to prove that implies . Moreover, prove that if then gives a well-defined morphism.


In geneneral, it's helpful to remember that as abelian groups .

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)?
how about 1-->3?
does 1-->2 share these problems? 1-->0?

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or, one could ignore the "element" view, and just think about what we know about homomorphisms. f(Z/2Z) has to be a subgroup of Z/4Z. how many subgroups of Z/4Z exist? is it even possible for f to be surjective?

also, ker(f) has to be a subgroup of Z/2Z. again, what are our possible choices? can you establish some kind of relationship between possible subgroups ker(f) of Z/2Z, and subgroups f(Z/2Z) of Z/4Z?