A discrete normal subgroup is contained in the center

Question

Let $G$ be a connected Lie group and $N$ a discrete normal subgroup of $G$. Then $N$ is contained in the center $Z(G)$.

I've fooled around with this for a little bit and I can't figure out how to use the hypothesis that $N$ is discrete. I think maybe it uses some fact that I do not know.

Answer 1

Let $n$ be an element of the subgroup. Then the function $f:G\to N$ sending $g\mapsto gng^{-1}$ is continuous, and since $G$ is connected, so is the image of $f$. Since $N$ is discrete, $f(g) =n$ for all $g$.