Exercise 7.11 in Fulton's Representation Theory asks to prove that:
(a) Show that any discrete normal subgroup of a connected Lie group $G$ is in the center $Z(G)$
(b) If $Z(G)$ is discrete, show that $G/Z(G)$ has trivial center.
I was able to solve (a) relatively easily (for $y \in N$, with $N$ a normal discrete subgroup of $G$) by considering the continuous mapping $\phi_y : G \to G$ for $ g \mapsto gyg^{-1}y^{-1}$. Since the map is continuous and $G$ is connected, its image must be connected. But since the image is contained in $N$ which is discrete, then the image must be trivial, implying that $gy = yg$ so $y \in Z(G)$ so $N \subset Z(G)$.
However, I am unsure of how to proceed in (b). I know that $G/Z(G)$ is the same as the group of inner automorphisms of $G$, but I don't know if that's how I should proceed or if I should take a different tactic.
Thanks for any help with this problem.
Later Edit:
Since I have to turn this in shortly, I want to say that I eventually ended up using a result from Stillwell's "Naive Lie Theory" that states that $Z(G)$ discrete implies that there are no nondiscrete normal subgroups. Thus, it is pretty easy to show that $G/Z(G)$ is simple, implying that its center must be trivial. However, Stillwell's result uses machinery that won't be introduced in Fulton's text for quite a while, so I'm still unsatisfied with the result. I eventually decided there were at least three possible approaches to the problem:
- What I ultimately did, except actually using the machinery available up to this point in Fulton to prove Stillwell's result and proceeding from there
- Possibly something involving $G/Z(G) \cong Inn(G)$, though I still don't know what
- Proving that $G=[G,G]$ (the commutator of $G$), then using Grun's Lemma to immediately show that the center of $G/Z(G)$ is trivial, though I'm not sure that having $Z(G)$ discrete even implies this fact
I still would like to know a decent solution, so any help/suggestions/proofs would still be more than welcome.