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What is the motivation/intuition behind these concepts? What notion/property of a group do they capture? Or what is the scenario of application.

Thanks.

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The centralizer of an element and the normalizer of a subgroup are both special cases of the stabilizer of a point for a group action. When $G$ acts on $G$ by conjugation, the stabilizer of a point in $G$ is its centralizer in $G$. When $G$ acts on its subgroups by conjugation, the stabilizer of a subgroup $H$ is its normalizer in $G$. Applications: the size of the conjugacy class of $g$ is the index of the centralizer of $g$, and the number of subgroups of $G$ that are conjugate to $H$ is the index of the normalizer of $H$ in $G$. Normalizers of Sylow subgroups appear in the 3rd Sylow thm. –  KCd Dec 23 '12 at 6:46
    
math.stackexchange.com/a/215146/12952 Check out my answer here. –  Alexander Gruber Dec 23 '12 at 21:55
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I've only recently studied group theory, so take this with a grain of salt, but my intuition is as follows: given group $G$ and subset $A \subset G$, the centralizer of $A$ "measures how 'inside' the center $Z(G)$ the set is" and the normalizer $A$ measures "how normal the set is". So if we have $C_G(A) = G$, then every element of $A$ commutes with every element of $G$ and hence $A \subset Z(G)$. Similarly if we have $N_G(A) = G$ then for any $g \in G, gA = Ag$ and hence $A$ is normal in $G$.

One common application that I have seen in my (again, limited) exposure uses the fact that, for subgroup $H \leq G, N_G(H)/C_G(H) \cong$ a subgroup of $Aut(H)$ (the group of automorphisms on $H$). Often depending on the identity of $H$ we can get a lot of information about possible subgroups of the automorphism group, and if $H$ is normal in $G$ then we have $G/C_G(H) \cong$ a subgroup of $Aut(H)$, and this tells us even more. This sort of approach is often helpful for proving that $H$ is contained in $Z(G)$.

As far as how the Stabilizer connects to these things, notice that if we let $G$ act on $H \leq G$ by conjugation (i.e. $g \cdot H = gHg^{-1}$) then $Stab(H) = N_G(H)$. This ends up leading into the Class equation and other useful theorems.

Hopefully that helped somewhat?

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@KCd Thanks. The explanation here seems to offer the kind of geometric intuition I was looking for. I need to work out more to fully grasp these ideas. –  hyper_w Dec 23 '12 at 15:23
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I see things slightly differently. Saying the centralizer measures proximity or intersection with the center is not true. What the centralizer does is it constructs a largest subgroup with the property that its center contains the given set. Likewise the normalizer constructs a largest subgroup for which the set is invariant under conjugation. Overall I think both are much more than just a measure of a certain property. –  peoplepower Dec 24 '12 at 0:47
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