This is probably stupid question, but I can't see the difference between the two subgroups:
$$C_G(A)=\{g\in G| gag^{-1}=a,\forall a\in A\}$$
$$Z(G)=\{g\in G| ga=ag,\forall a\in G\}$$
Is the difference that the centralizer takes a subset $A\in G$ and the center always uses the entire group $G$?
Best Answer
Converting the comments into an actual answer:
Yes, we have the special case $C_G(G)=Z(G)$, so the notion of centralizer can be thought of as a generalization of centers, since the centralizer $C_G(A)$ works for every subset $A$ of $G$ (note that it doesn't have to be a subgroup).
As for the normalizer, $Z(G)\subseteq C_G(A)\subseteq N_G(A)$ for every subgroup $A\leq G$, and $N_G(A)$ is the maximal subgroup of $G$ in which $A$ is normal. The normalizer can also be defined for an arbitrary subset, and the above still holds other than that we must now say that $\langle A \rangle$, the subgroup generated by $A$, is normal in $N_G(A)$, and the normalizer is maximal with respect to this property.