Injective and surjective homomorphisms between non cyclic group of order $4 $ to $Z_8$ 
Let $G$ be a non cyclic group of order $4$. Consider the following
  statements:
$I:$ There is no one-one map homomorphism from $G$ to $Z_8$
$II:$ There is no onto homomorphism from $Z_8$ to $G$
Then which of these statements are true?

Okay, So by classification, $G\simeq Z_2\times Z_2$
Consider statement $I$
If $\phi $ is $1-1$ homomorphism $\phi:G\to^{1-1}\to Z_8\rightarrow \frac{G}{ker\phi}\simeq H \leq Z_8$, where $H$ is a subgroup of $Z_8$ and hence cyclic. But here $ker\phi = \{e\}$ so,
$G\simeq H \leq Z_8$ which is not possible since $G$ is not cyclic.
So $I$ is true.
Consider statement $II$
if $\phi$ is onto homomorphism from $\phi: Z_8\to G$, then $\frac{Z_8}{ker\phi}\simeq G$
But $\frac{Z_8}{ker\phi}$ is cyclic since $Z_8$ is cyclic, so $\frac{Z_8}{ker\phi}\simeq G$ is not possible since $G$ is non cyclic.
 So $II$ is true.
So both statements are true. Is this correct?

I think your arguments are correct, but can be shortened. 
Part I: Suppose that there is such a homomorphism $\phi$. Then
$\phi(G)\cong G$ is a subgroup of $C_8$, hence cyclic, a contradiction.
Part II: Suppose that $\phi$ is such a homomorphism. Then $\phi(C_8)\cong G$, where $\phi(C_8)$ is cyclic, because $C_8$ is cyclic, a contradiction.
