Describing homomorphisms, kernals and ranges The question states:
Describe all homomorphisms $f:\mathbb Z_4 \to \mathbb Z_8$. Describe their kernals and ranges.
What I have so far:
There are GCD$(4,8)$ homomorphisms from $\mathbb Z_4$ to $\mathbb Z_8$, therefore there are $4$ homomorphisms.
$\mathbb Z_4,8=\langle 1\rangle$ implies $f(1)=n$ for any $n \in \mathbb Z_8$
so $f(k)=f(k*1)=kf(1)=kn$ for all $k \in \mathbb Z_4$.
I'm confused about how to use this to find the homomorphisms though.The same goes for Ker$(f)$ and Ran$(f)$.
I appreciate any help!
 A: Since $\mathbb{Z}_8$ is cyclic, every subgroup of it is cyclic as well. (Can you show this?)
Consider where $1 \in \mathbb{Z}_4$ could be mapped to in $\mathbb{Z}_8$. For each possibility, figure out what the resulting kernels and ranges would be.
Example: Suppose our homomorphism $f: \mathbb{Z}_4 \rightarrow \mathbb{Z}_8$ sends $1 \mapsto 4$.
Then:
$f(1) = 4$; 
$f(2) = f(1+1) = f(1) + f(1) = 4+4 = 0$;
$f(3) = f(1+1+1) = f(1) + f(1) + f(1) = 4+4+4 = 4$;
and $f(0) = f(1+1+1+1) = f(1) + f(1) + f(1) + f(1) = 4+4+4+4 = 0$,
where we have used the fact that $f$ is a homomorphism to break up our sums mod $8$.
To conclude: $1, 3 \mapsto 4$ and $0, 2 \mapsto 0$, so we end up with a two element group.
In particular, we have $\{0, 4\}$ under addition modulo $8$, which is isomorphic to $\mathbb{Z}_2$.
As you can see, the range is $\{0, 4\}$ and the kernel (the elements sent to $0$) is $\{0, 2\}$.
A: Notice that $f$ is an epimorphism on to $Ran(f)$ so $Ran(f)$ has to be isomorphic to $0$, $\mathbb Z_2$ or $\mathbb Z_4$. $Ran(f)$ is a subgroup of $\mathbb Z_8$. Now you just have to find all such subgroups of $\mathbb Z_8$.
A: Well, as you've noticed, for any homomorphism $f:\Bbb Z_4\to\Bbb Z_8$ and any $k,$ we have $f(k)=k\cdot f(1).$ Such a homomorphism will be well-defined if and only if $f(5)=f(1).$ (Why?)
For which $n\in\Bbb Z_8$ do we have $n=5n$? These are the $n$ for which we can uniquely define a homomorphism $f_n:\Bbb Z_4\to\Bbb Z_8$ by $f_n(k)=kn,$ and there are only $4$ such $n$. Once you figure out which $n$ work, it will be easier to find the kernel and image for each $f_n$.
