Difference between Stabilizer and Centralizer? I know that the Centralizer of an element $a$ in a group $G$ is defined as follows $$C_G(a) = \{ g \in G \space  | \space ga = ag \}.$$
It can also be defined as follows
$$C_G(a) = \{ g \in G \space  | \space  gag^{-1} = a \}$$ 
Informally, It contains all the elements in $G$ that commutes with $a$
And also I know that $C_G(a)$ is a subgroup of $G$
Now the center of a group is defined as follows 
$$Z(G) = \{ g \in G \space | \space gx =xg \space\forall x \in G \}$$
I also know that the center $Z(G)$ of the group $G$ is a subgroup of $G$ and $Z(G) \subseteq C_G(a)$ and hence $Z(G)$ is a subgroup of $C_G(a)$
And so $Z(G) \leq C_G(a) \leq G $ where $\leq$ denotes a subgroup
Now I am very confused about the nature of the Stabilizer.
All I know is that if $G$ is a group acting on a set $S$ and $s \in S$ then the stabilizer of $s$ in $G$ is the set $$G_s = \{ g\in G \space | \space gs = s \}.$$
Here are my questions
(1) Is the stabilizer a subgroup of $G$
I figured this one out. First of all it's not empty since by the definition of a group action $es = s$ and so $e \in G_s$
Now let $g,h \in G_s$ then $(gh)s = g(hs)= g(s) =s$ hence $gh \in G_s$
Now let $g \in G_s$ then $g^{-1}s = g^{-1} (gs) = (g^{-1}g)s = es = s$ and so $g^{-1} \in G_s$ and hence $G_s \leq G$
(2) What is the difference between the Stabilizer and the centralizer because most of the time I see them used interchangeably especially when dealing with the number of element in a conjugacy class.
(3) Is it true that $[G:C_G(a)] = [G:G_a]$?
(4) Where does that stabilizer fit in  here  $Z(G) \leq C_G(a) \leq G $ where $\leq$  denotes subgroup?
 A: As a preliminary, you cannot talk about stabilizer without specifying an action.
(1) Yes.
(2)The centralizer in $G$ of $a\in G$ is the stabilizer of $a$ for $G$ acting on $G$ by conjugation (i.e. $g.h:=ghg^{-1}$). So the centralizer may be a stabilizer, it actually depends on the action.
(3) Yes, if you choose the good action.
(4) If you are still considering the action of $G$ on $G$ by conjugation then $Z(G)$ is the kernel of the action (it is also the globally fixed points set). This implies ( because each stabilizer contains the kernel of the action) $Z(G)\subseteq C_G(a)$.
A: You should start off by answering question #1 yourself; this is a great collection of questions, and you're definitely capable of it.
Every group $G$ acts on itself by conjugation; $x \mapsto x^g = g^{-1}xg\ $ (prove this).  Under this action, you should figure out what the more common names for "orbit" and "stabilizer" of a group element are. This will show you how centralizers and stabilizers are related. Note that the relationship is that of "proper containment", as there are other ways a group can act on itself besides conjugation (although conjugation is a very special action). This should also shed more light on question #3, if you know a certain famous theorem about group actions.
For bonus fun, let $\cal{S}$ be the set of subgroups of $G$. See what happens when $G$ acts on $\cal{S}$ by conjugation; do stabilizers or orbits of subgroups have any more common names? 
