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Abstract Algebra

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Normalizers Versus Centralizers

ABSALG-ZWV@HX

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 $N(H)=\{g\in G:gHg^{-1}=H\}$. The centralizer of $H$ in $G$ is denoted by $C_G(H)$ or $C(G)$, and is defined as $C(H)=\{g\in G: gh=hg\; \text{for all}\; h\in H\}$.

A group homomorphism from a group $G$ to itself that is bijective is called an automorphism of $G$. For every fixed element $g\in G$, conjugation by $g$, that is $x\mapsto gxg^{-1}$ defines an automorphism of $G$. Each such automorphism is called an inner automorphism of $G$.

Which of the following facts is NOT true about the normalizer and centralizer? Select ALL that apply.

A

$N(H)=C(H)$.

B

$C(H)$ is a normal subgroup of $N(H)$.

C

$N(H)/C(H)\cong \text{Inn}(H)$, where $\text{Inn}(H)$ is the group of inner automorphisms of $H$.

D

$H\subseteq C(C(H))$.

E

$N(H)/C(H)\cong \text{Aut}(H)$, where $\text{Aut}(H)$ is the group of automorphisms of $H$.