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The normalizer $N_G(H)=\{g\in G:gHg^{-1}=H\}$ of a subgroup

Definition

Let $H\le G$ be a subgroup (Subgroup). The normalizer of $H$ in $G$ is

$N_G(H):=\{g\in G:gHg^{-1}=H\}.$

Thus $g\in N_G(H)$ exactly when the conjugation automorphism $c_g$ preserves $H$ setwise (Conjugation $x\mapsto gxg^{-1}$ is an automorphism). The subgroup property is proved in $C_G(x)$ and $N_G(H)$ are subgroups of $G$ ↗.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets (standard reference, not scraped)

The normalizer NG(H)={g∈G:gHg−1=H}N_G(H)=\{g\in G:gHg^{-1}=H\}NG​(H)={g∈G:gHg−1=H} of a subgroup