Difference between a group normalizer and centralizer

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 $N_G(A)$?

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: $$C_G(A) \mbox{ is always contained in } N_G(A).$$

Consider $G$ acting by conjugation on itself. Elements of $N_G(A)$ restrict to permutations of $A$, and more particularly elements of $C_G(A)$ restrict to the identity permutation of $A$.

Let H is a Subgroup of G. Now if H is not normal if any element ${g \in G}$ doesn't commute with H. Now we want to find if not all ${g \in G}$, then which are the elements of G that commute with every element of H? they are normalizer of H. i.e., the elements of G that vote 'yes' for H when asked to commute.

• Hence, ${N_G(H)=\{g \in G: gH=Hg }\}$
• | Now Centralizer of an element ${a \in G}$ is set of all elements that commutes with ${a}$. in layman's term element ${a}$ looks for the element in G that commutes with it.
• Hence ${C(a)=\{g\in G : ag=ga}\}$