If a homomorphism is surjective it cannot be trivial

Question

My group theory is a bit rusty, but I recall such a property existing. Is this true, if so, can anyone give me some hints as to how to prove this?

Answer 1

You need an additional hypothesis. If a homomorphism whose codomain is a nontrivial group is surjective then it is nontrivial (this is immediate from the fact that the trivial homomorphism by definition takes on only one value), but of course the trivial homomorphism to the trivial group is surjective.

Answer 2

Let $G$ be a group. Consider

$$\begin{align} \varphi: G &\to \{e\}\\ g & \mapsto e. \end{align}$$

Then for any $g,h\in G$, we have $\varphi (gh)=e=ee=\varphi(g)\varphi(h)$, so $\varphi$ is a homomorphism.

Also, we have, for $e\in \{e\}$, there exists a $k\in G$ such that $e=\varphi(k)$ by definition of $\varphi$; namely, we can take $e_G=k$. So $\varphi$ is surjective.

But, clearly, $\varphi$ is trivial.