Normalizer of Center is Group

Theorem
Let $G$ be a group.

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

Let $x \in G$.

Let $\map {N_G} x$ be the normalizer of $x$ in $G$.

Then:
 * $\map Z G = \set {x \in G: \map {N_G} x = G}$

That is, the center of a group $G$ is the set of elements $x$ of $G$ such that the normalizer of $x$ is the whole of $G$.

Thus:
 * $x \in \map Z G \iff \map {N_G} x = G$

and so:
 * $\index G {\map {N_G} x} = 1$

where $\index G {\map {N_G} x}$ is the index of $\map {N_G} x$ in $G$.

Proof
$\map {N_G} x$ is the normalizer of the set $\set x$.

Thus: