Let $G$ and $H$ be two groups. Suppose I'm able to define a non trivial group homomorphism from $G$ to $H$ what does it actually tell? Is it like $G$ is similar to $H$ in some sense?
Also further I may or may not be able to define a non trivial group homomorphism from $H$ to $G$. Suppose I'm unable to define then from my first conclusion and this can I say $H$ is not similar to $G$ in some sense?
Is my idea correct?
On
Let $\phi\colon G\rightarrow H$ be a group homomorphism. Then it depends on the other properties of $\phi$ how "similar" the groups $G$ and $H$ are. Of course, the idea is that $\phi$ is injective and surjective, so that $G$ and $H$ can be viewed as the same group. We say that $G$ and $H$ are isomorphic then. If $\phi$ is injective, we can identify $G$ as a subgroup of $H$. If $\phi $ is surjective, then $H$ is a homomorphic image of $G$ and inherits several properties of $G$, like solvability, for example. In general, we can only say that $$ G/\ker(\phi)\cong \phi(G)\le H. $$
I don't understand what you mean by “similar” here, but I assuming that, for you, if $G$ is similar to $H$, then $H$ is similar to $G$, then the answer is negative. In fact, there is a non-trivial group homomorphism from $\Bbb Z$ into $\Bbb Z_2$, but there is no non-trivial group homomorphism from $\Bbb Z_2$ into $\Bbb Z$.