Centralizer, Normalizer and Stabilizer - intuition 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.
 A: 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?
