Central Subgroup is Normal

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Theorem

Let $G$ be a group.

Let $H$ be a central subgroup of $G$.


Then $H$ is a normal subgroup of $G$.


Proof 1

Let $H$ be a central subgroup of $G$.

By definition of central subgroup:

$H \subseteq \map Z G$

where $\map Z G$ is the center of $G$.


Thus we have that $H$ is a group which is a subset of $\map Z G$.

Therefore by definition $H$ is a subgroup of $\map Z G$.

We also have from Center of Group is Abelian Subgroup that $\map Z G$ is an abelian group.

It follows from Subgroup of Abelian Group is Normal that $Z$ is a normal subgroup of $G$.

$\blacksquare$


Proof 2

Let $H$ be a central subgroup of $G$.

By definition of central subgroup:

$H \subseteq \map Z G$

where $\map Z G$ is the center of $G$.


Then:

\(\displaystyle \forall x \in G: \forall h \in H: \ \ \) \(\displaystyle x h x^{-1}\) \(=\) \(\displaystyle x x^{-1} h\) as $h \in H \implies h \in \map Z G$
\(\displaystyle \) \(=\) \(\displaystyle h\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x h x^{-1}\) \(\in\) \(\displaystyle H\) as $h \in H$
\(\displaystyle \leadsto \ \ \) \(\displaystyle H\) \(\lhd\) \(\displaystyle G\) Definition of Normal Subgroup

$\blacksquare$


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