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Abstract Algebra

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Moderate

Characterizing the Normalizer of Subgroups

ABSALG-R1J9ZB

Recall that for every subgroup $H$ of a group $G$, the normalizer of $H$ in $G$ is denoted by $N_G(H)$, or simply $N(H)$ and is defined as $N(H)=\{g\in G:gHg^{-1}=H\}$.

Which of the following facts is true about the normalizer?

A

$N(H)=G$ if and only if $H$ is normal in $G$.

B

For all subgroups $H_1, H_2$ of a group $G$. If $H_1\subseteq H_2$, then $N(H_1)\subseteq N(H_2)$.

C

For all subgroups $H_1, H_2$ of a group $G$. If $H_1\subseteq H_2$, then $N(H_2)\subseteq N(H_1)$.

D

Normalizers of distinct subgroups are distinct.