Center is Characteristic Subgroup

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Theorem

Let $G$ be a group.


Then its center $\map Z G$ is characteristic in $G$.


Proof

Let $\phi$ be an automorphism of $G$.

Let $x \in \map Z G, y \in G$.

Then:

\(\displaystyle \map \phi x y\) \(=\) \(\displaystyle \map \phi x \map \phi {\map {\phi^{-1} } y}\) automorphisms are bijections
\(\displaystyle \) \(=\) \(\displaystyle \map \phi {x \map {\phi^{-1} } y}\) Definition of Group Homomorphism
\(\displaystyle \) \(=\) \(\displaystyle \map \phi {\map {\phi^{-1} } y x}\) Definition of Center of Group
\(\displaystyle \) \(=\) \(\displaystyle \map \phi {\map {\phi^{-1} } y} \map \phi x\) Definition of Group Homomorphism
\(\displaystyle \) \(=\) \(\displaystyle y \map \phi x\)

Hence $\map \phi x \in \map Z G$.

So we have $\phi \sqbrk {\map Z G} \subseteq \map Z G$.

Since $\phi^{-1}$ is also an automorphism:

$\phi^{-1} \sqbrk {\map Z G} \subseteq \map Z G$

Since $\phi$ is a bijection:

$\map Z G = \phi \sqbrk {\phi^{-1} \sqbrk {\map Z G}} \subseteq \phi \sqbrk {\map Z G}$


Therefore we conclude that $\phi \sqbrk {\map Z G} = \map Z G$.

Hence $\map Z G$ is characteristic in $G$.

$\blacksquare$