Can the pre-image (under homomorphism) of a subgroup be empty? I'm asked to prove that if $E \leq H, \varphi^{-1}(E) \leq G$ where $\varphi: G \rightarrow H$ is an homomorphism.
I can show that $\varphi^{-1}(E)$ satisfies the group condition of $\forall x, y \in \varphi^{-1}(E), xy^{-1} \in \varphi^{-1}(E)$. 
But how do I know that $\varphi^{-1}(E) \neq \emptyset$? I don't see why this needs to be true if $\varphi$ is not an isomorphism.

Grouphomomorphism $\phi$ sends $\mathsf{id}_G$ to $\mathsf{id}_H\in E$ so $\mathsf{id}_G\in\phi^{-1}(E)$.
