If a group centralizer is defined as $C_G(A)=\{g \in G : gag^{-1} = a$ for all $a \in A\}$, and a group normalizer is defined as $N_G(A)=\{g\in G:gAg^{-1}=A\}$, where $gAg^{-1}=\{gag^{-1}:a\in A\}$ (definition taken from Abstract Algebra by Dummit and Foote), then what's the difference between $C_G(A)$ and $N_G(A)$?
[Math] Difference between a group normalizer and centralizer
abstract-algebragroup-theory
Best Answer
I would say (less precisely, but correctly) like this:
$g$ is in $N_G(A)$ means $gag^{-1}=$ some $a'$ in A ($a\in A$).
$g$ is in $C_G(A)$ means $gag^{-1}=$ same $a$ in A ($a\in A$).
We should note that although there is difference between these two notions, there is also a relation between them: $$C_G(A) \mbox{ is always contained in } N_G(A).$$