TITLE: Number of preimages of a group element under a homomorphism.
QUESTION [5 upvotes]: I was reading the first chapter of Robert Gilmore's "Lie groups, physics and geometry" and I came across a brief statement regarding the number of preimages of an element under an homomorphism which I can't seem to prove. Here's the whole paragraph:
"The group $H$ is called a homomorphic image of $G$. Several different group elements in $G$ may map to a single group element $H$. Every element $h_i \in H $ has the same number of inverse images $g_i \in G$."
Anyone knows why is it that this is the case? I've been trying to construct a proof but getting nowhere.
Thanks in advance.

REPLY [4 votes]: Let $\varphi$ be the homomorphism, let $1_H$ be the identity element of $H$, and let $N=\varphi^{-1}(1_H)$. If $g\in G$ and $\varphi(g)=h\in H$ then $\varphi^{-1}(h)=Ng$, whence $|\varphi^{-1}(h)|=|N|$.