Discrete Normal Subgroup of Connected Group is Contained in Center

Theorem
Let $G$ be a connected topological group.

Let $\map Z G$ be the center of $G$.

Let $N$ be a discrete normal subgroup of $G$.

Then:
 * $N \subseteq \map Z G$

Proof
Let $h \in N$.

We shall show $h \in \map Z G$

Now, let:
 * $A _h := \set {g \in G : g^{-1} h g = h}$

Then, we need to show:
 * $A _h = G$

Let $e \in G$ denote the identity.

Since $e \in A _h$, we have $A _h \ne \O$.

Thus, it suffices to show that $A _h$ is clopen in view of.

As $N$ is a discrete subgroup, there is an open set $U \subseteq G$ such that:
 * $U \cap N = \set e$

For each $h \in N$, multiplying the above equality by $h$ from left, we obtain:
 * $h U \cap N = \set h$

Define $\phi _h : G \to N$ by:
 * $\map {\phi _h} g := g^{-1} h g$

Then $\phi _h$ is continuous.

Thus both:

and

are open.

That is, $A _h$ is clopen.