Normalizer of Center is Group

From ProofWiki

Jump to: navigation, search

Theorem

Let $G$ be a group.

Let $Z \left({G}\right)$ be the center of $G$.

Let $x \in G$.

Let $N_G \left({x}\right)$ be the normalizer of $x$ in $G$.

Let $\left[{G : N_G \left({x}\right)}\right]$ be the index of $N_G \left({x}\right)$ in $G$.

Then:

$Z \left({G}\right) = \left\{{x \in G: N_G \left({x}\right) = G}\right\}$

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 $x \in Z \left({G}\right) \iff N_G \left({x}\right) = G$, and so $\left[{G : N_G \left({x}\right)}\right] = 1$.

$N_G \left({x}\right)$ is the normalizer of the set $\left\{{x}\right\}$. Thus:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle N_G \left({x}\right)$	$=$	$\displaystyle G$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \forall a \in G: \left\{ {x}\right\}^a$	$=$	$\displaystyle \left\{ {x}\right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Normalizer
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \forall a \in G: a x a^{-1}$	$=$	$\displaystyle x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Conjugate
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \forall a \in G: a x$	$=$	$\displaystyle x a$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle Z \left({G}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Center of a Group

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle N_G \left({x}\right)\)	\(=\)	\(\displaystyle G\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \forall a \in G: \left\{ {x}\right\}^a\)	\(=\)	\(\displaystyle \left\{ {x}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Normalizer
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \forall a \in G: a x a^{-1}\)	\(=\)	\(\displaystyle x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Conjugate
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \forall a \in G: a x\)	\(=\)	\(\displaystyle x a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle Z \left({G}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Center of a Group