Difference between centralizer and center groups?

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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 $G$?

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2 Answers

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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.

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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.

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