I am a little confused when working with the inverse of a homomorphism, this specifically came up when proving the following claim:
Let $\phi$ be some homomorphism from $G_1\rightarrow G_2$: $$H_2 \subset G_2\textrm{ is a normal subgroup } \Rightarrow \phi^{-1}(H_2)\subset G_1 \textrm{ is a normal subgroup}$$
The proof for this begins by stating that $$e_1\in\phi^{-1}(H_2)\textrm{ since } \phi(e_1)=e_2\in H_2$$ But since $\phi$ is a homorphism, it is not nessesarily surjective or injective. So there could be many terms in $g_1,g_2,...\in G_1$ such that $\phi(g_1)=\phi(g_2)=...=e_1$. So does $\phi^{-1}(e_1)$ return all of these values? Can the inverse function of a Homomorphism return multiple values per input? In general, when dealing with inverse functions, should I think of the inverse as a function that will return every coresponding value?
The expression $\phi^{-1}(H_2)$ means the set $\{g\in H_1\mid\phi(g)\in H_2\}$, for any function $\phi$, even if it has no inverse.