TITLE: Difference between centralizer and center groups?
QUESTION [22 upvotes]: 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 $G$?

REPLY [5 votes]: The normalizer of a subset of a group is the part of the group that commutes with the subset (but not necessarily holding the element of the subset fixed): $g$ in the normalizer of $H \subset G$ means $gH = Hg$, or more specifically, $gh_1 = h_2 g$ for some $h_1, h_2 \in H$.
The centralizer of a subset of a group is the part of the group that commutes with the subset elementwise (i.e., holding the element of the subset fixed): $g$ in the centralizer of $H \subset G$ means $gh = hg$ for all $h \in H$.  Clearly, commuting elementwise implies commuting with the subset, so the normalizer of a subset contains the centralizer of that subset.
The center of a group is the part of the group that commutes with everything in the group.  Commuting with everything implies commuting with elements of some subset, so the centralizer of a subset contains the center of the group.
Putting these together: $Z(G) \subset C_G(H) \subset N_G(H)$.
Regarding "$\leq$"...  The centralizer and normalizer are defined on subsets.  It may turn out that a subset is also a subgroup, but this is not required.  (It will essentially always be the case that the subsets are subgroups once you are past the material introducing these definitions.)  This is why I have used "$\subset$" in the above.