The following has appeared in page 63 of Combinatorial Group Theory notes by Charles. F. Miller.
I am trying to understand the proof of the following theorem.
Theorem(Malcev): Finitely generated residually finite groups are Hopfian.
In part of the proof says the following, which I can't understand:
"Suppose that $G$ is finitely generated and residually finite. Let $\psi:G\rightarrow G$ be an epimorphism. Let $H$ be a subgroup of finite index, say $n=[G:H]$. Then $\psi^{-1}(H)$ is again a subgroup of finite index $n$ in $G$..."
I understand the fact that $\psi^{-1}$ is a subgrgroup of $G$. But why $[G:\psi^{-1}(H)]=n$?