How to determine the total number of homomorphisms from $V_4 \rightarrow C_2$? Where $V_4$ is the Klein-$4$ group and $C_2$ is cyclic of order $2$.
So far I have managed to create $7$ just from playing about with the elements.
These are: The trivial homomorphism. $3$ homomorphisms in which all elements of $V_4$ are mapped to the identity in $C_2$ except one non identity element which is mapped to the non identity in $C_2$. And finally the $3$ homomorphisms taking $2$ distinct non-identity elements from $V_4$ and mapping them to the non identity in $C_2$.
I have a few questions.
1) If $\phi:G \rightarrow H$ is a homomorphism what can we say about the order of elements? (I.e. is it true that $o(g)|o(\phi(g))$ or the other way around maybe?)
2) Are my maps correct and are the the only homomorphisms from $V_4$ to $C_2$.
3) Is there a nicer way to argue this question other than just trying to guess all possible ones?