How to prove that there is not a monomorphism from Klein 4-group to $Z_6$(or a epimorphism from $Z_6$ to $V_4$)?

How to prove that there is not a monomorphism from Klein 4-group to $Z_6$(or a epimorphism from $Z_6$ to $V_4$) ?

I know that:
If $f$ is a monomorphism than $Kernel(f)={0}$ .

In $V_4$ , three elements have order 2.

In $Z_6$ , $’3’$ has order 2, $’2’$ and $’4’$ have order 3 , $’1’$ and $’5’$ have order 6.

I read a theorem which says:"If $G$ and $G’$ are finite groups and $f$ is a homomorphism between them so that $f(a)=a’$ , where $a$ is from $G$ and $a’$ is from $G’$ , then the order of $a’$ divides the order of $a$."

My question is:Using this theorem, can I say that the monomorphism between $V_4$ and $Z_6$ does not exist since orders $3$ and $5$ do not divide $2$(order of all elements of Klein group)?