Union of Conjugacy Classes is Normal

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Theorem

Let $G$ be a group.

Let $H \le G$.


Then $H$ is normal in $G$ if and only if $H$ is a union of conjugacy classes of $G$.


Proof

\(\ds H\) \(\lhd\) \(\ds G\) where $\lhd$ denotes that $H$ is normal in $G$
\(\ds \leadstoandfrom \ \ \) \(\ds \forall g \in G: \, \) \(\ds g H g^{-1}\) \(\subseteq\) \(\ds H\) Definition of Normal Subgroup
\(\ds \leadstoandfrom \ \ \) \(\ds \forall x \in H: \forall g \in G: \, \) \(\ds g x g^{-1}\) \(\in\) \(\ds H\)
\(\ds \leadstoandfrom \ \ \) \(\ds \forall x \in H: \, \) \(\ds \conjclass x\) \(\subseteq\) \(\ds H\) where $\conjclass x$ is the conjugacy class of $x \in G$
\(\ds \leadstoandfrom \ \ \) \(\ds H\) \(=\) \(\ds \bigcup_{x \mathop \in H} \conjclass x\)

Hence the result.

$\blacksquare$


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