Center of Group is Normal Subgroup
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Theorem
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The center $\map Z G$ of any group $G$ is a normal subgroup of $G$ which is abelian.
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
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $10$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.11 \ \text{(a)}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46 \theta$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $5$