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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?

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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?

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  • $\begingroup$ I answered the 1st question. Yes It is not that hard. $\endgroup$ – alkabary Mar 31 '15 at 21:21
  • $\begingroup$ @leo Nice! I know what I wrote is very cryptic, but I'm sure that if you just unpack the definitions of $\operatorname{Orb}_G(x)$ and $\operatorname{Stab}_G(x)$ for $x \in G$ under the conjugation action, you'll be able to see things you find very familiar (at least I learned their other names before learning about group actions). Making the connection is much more fun than being told the connection :) $\endgroup$ – pjs36 Mar 31 '15 at 21:34
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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) $Z(G)\subseteq C_G(a)$.

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