# Surjective group homomorphism between quotient groups

Assumptions: Assume that $G$ is a topological group and $Z_1,Z_2$ are discrete, normal subgroups of $G$ (hence central) and $G / Z_1$ and $G / Z_2$ denote the quotient groups. Assume moreover that there exists a group homomorphism $\varphi \colon G / Z_1 \to G / Z_2$, which is also a topological covering map.

Claim: Then $Z_1 \subseteq Z_2$.

Idea: Of course we know that $\varphi(e \cdot Z_1) = e \cdot Z_2$. But since we have not made any assumptions about $\varphi$ respecting the natural action of $G$ on both spaces, I don't know how to proceed further. Maybe I don't even need that $\varphi$ is a covering map, just the fact that $\varphi$ is surjective.

• As stated this is false: take $Z_1=2\pi\mathbf Z$ and $Z_2=\mathbf Z$ in $\mathbf R$. Then there is even a homeomorphism from $\mathbf R/Z_1$ to $\mathbf R/Z_2$, but $Z_1\not\subseteq Z_2$. – Johannes Huisman Jun 3 '16 at 14:23
• Ok, this is bad and good. :) But now I know I need some additional structure. – varsop Jun 3 '16 at 14:40