Centralizer subgroup: $$C_G(A)=\{g\in G\mid gag^{-1}=a,\forall a\in A\}$$ Center subgroup: $$Z(G)=\{g\in G\mid ga=ag,\forall a\in G\}$$
Centralizer and center are subgroups of $G$. The centralizer takes a subset $A\in G$ and the center always uses the entire group $G$. So the centralizer becomes center when $A=G$ $$ C_G(G)= Z(G). $$
My questions are that:
Under what precise mathematical conditions that the normalizer $N_G(A)$ equal to a normal subgroup $N$ of $G$? When is $N_G(A)$ not equal to a normal subgroup $N$? See for instances.
Since the quotient group is the $G/N$, are there analogous concepts of quotient (not a subgroup) for $G/N_G(A)$, important for what properties or theorems?
In short, I am exhausting the concepts of
Centralizer : center subgroup ~ normalizer : normal subgroup ~ ?: quotient group