Center of Group is Normal Subgroup

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Theorem

Let $G$ be a group

The center $\map Z G$ of $G$ is a normal subgroup of $G$.


Proof 1

Recall that Center of Group is Abelian Subgroup.

Since $g x = x g$ for each $g \in G$ and $x \in \map Z G$:

$g \map Z G = \map Z G g$

Thus:

$\map Z G \lhd G$

$\blacksquare$


Proof 2

We have:

$\forall a \in G: x \in \map Z G^a \iff a x a^{-1} = x a a^{-1} = x \in \map Z G$

Therefore:

$\forall a \in G: \map Z G^a = \map Z G$

and $\map Z G$ is a normal subgroup of $G$.

$\blacksquare$


Sources

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