# 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 |

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 11$