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.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $21$