Abstract Algebra

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Moderate

How Normal is the Normalizer?

ABSALG-S4TFZ3

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? Select ALL that apply.

A

$N(H)$ is normal in $G$.

B

$N(H)$ is contained in every normal non trivial subgroup of $G$.

C

$H$ is normal in $N(H)$.

D

$N(H)$ contains every subgroup of $G$ that has $H$ as a normal subgroup.