# Central Subgroup is Normal

## 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

$\blacksquare$