= Centralizer and normalizer
{wiki}
Both constructions are <stabilizer subgroups> for the <conjugation action>. The <centralizer> $C_G(x)$ fixes an individual element $x$; the <normalizer> $N_G(H)$ preserves a <subgroup> $H$ as a whole, while possibly moving its elements. The <centralizer of a subset> $C_G(H)$ fixes every member of $H$ individually, so
$$
C_G(H)\subseteq N_G(H).
$$
For a <cyclic subgroup> $H=\langle r\rangle$, commuting with the <group generator> $r$ is equivalent to commuting with all its powers, giving $C_G(H)=C_G(r)$.
In the <symmetric group> $S_3$, let $r=(123)$ and $H=\langle r\rangle=\{e,(123),(132)\}$. Then
$$
C_{S_3}(r)=C_{S_3}(H)=H,\qquad N_{S_3}(H)=S_3.
$$
Every <transposition> sends $r$ to $r^{-1}$ under <conjugation>. In particular, for $t=(12)$,
$$
trt^{-1}=r^{-1}\ne r,\qquad tHt^{-1}=\{e,r^{-1},r\}=H.
$$
Thus $t$ belongs to the <normalizer> of $H$ but not to the <centralizer> of $r$. Preserving the collection of powers allows the two possible directions of its <group generator> to be exchanged.
\Image[/s3-centralizer-versus-normalizer.png]
{title=The <centralizer> fixes $r=(123)$, while the <normalizer> preserves $H=\langle r\rangle$ and may exchange $r$ with its <inverse element>}
This distinction lets one ask separately which operations leave a chosen operation unchanged and which operations preserve the <subgroup> it generates. The <normalizer> also supplies a <group homomorphism> to the <automorphism group> of $H$, by $g\mapsto(h\mapsto ghg^{-1})$; its <group kernel> is $C_G(H)$. In this example the <group homomorphism> has two possible outputs, fixing $r$ or replacing it by $r^{-1}$. The <quotient group> $N_{S_3}(H)/C_{S_3}(H)$ is therefore a <cyclic group> of <group order> two. Milne develops the <normalizer> as a <stabilizer subgroup> in <Group Theory (v4.01) by James S. Milne>.
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