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Abstract Algebra

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Normality of Subgroups

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Let $G$ be a group. Identify which of the following subgroups $H$ of $G$ are normal in general. Select ALL that apply.

A

Center of $G$: $Z(G)=\{z\in G: zg=gz \textrm{ for all $g\in G$}\}$

B

Centralizer of a subset $S$ of $G$: $C_G(S)=\{g\in G: gs=sg \textrm{ for all $s\in S$}\}$

C

Normalizer of a subset $S$ of $G$: $N_G(S)=\{g\in G: gS=Sg \}$

D

Stabilizer of the element $x$ of $G$: $G_x=\{g\in G: gx=x\}$

E

None of the above.