Central Subgroup is Normal/Proof 2
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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
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:
\(\ds \forall x \in G: \forall h \in H: \, \) | \(\ds x h x^{-1}\) | \(=\) | \(\ds x x^{-1} h\) | as $h \in H \implies h \in \map Z G$ | ||||||||||
\(\ds \) | \(=\) | \(\ds h\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x h x^{-1}\) | \(\in\) | \(\ds H\) | as $h \in H$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds H\) | \(\lhd\) | \(\ds G\) | Definition of Normal Subgroup |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 49.5$ Normal subgroups