Normalizer and Centralizer of a Subgroup of Order 2
Problem 94
Let $H$ be a subgroup of order $2$. Let $N_G(H)$ be the normalizer of $H$ in $G$ and $C_G(H)$ be the centralizer of $H$ in $G$.
(a) Show that $N_G(H)=C_G(H)$.
(b) If $H$ is a normal subgroup of $G$, then show that $H$ is a subgroup of the center $Z(G)$ of $G$.
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Definitions.
Recall that the centralizer of $H$ in $G$ is
\[C_G(H)=\{g \in G \mid gh=hg \text{ for any } h\in H\}.\]
The normalizer of $H$ in $G$ is
\[N_G(H)=\{g\in G \mid gH=Hg \}.\]
Proof.
(a) Prove $N_G(H)=C_G(H)$
In general, we have $C_G(H) \subset N_G(H)$. We show that $N_G(H) \subset C_G(H)$.
Take any $g \in N_G(H)$. We have $gH=Hg$. Since $|H|=2$, let $H=\{1,h\}$.
Then $gH=\{g,gh\}$ and $Hg=\{g, hg\}$. Since $gH=Hg$, we have $gh=hg$. Namely $g\in C_G(H)$.
This proves that $N_G(H) \subset C_G(H)$, hence $N_G(H)= C_G(H)$.
(b) If $H$ is normal, then $H$ is a subgroup of $Z(G)$
Suppose that $H$ is a normal subgroup of $G$, that is $G=N_G(H)$.
By part (a), this implies that $G=C_G(H)$. Hence $H < Z(G)$.
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