# Concerning the point stabilizing group and coset stabilizing group.

I would like to know more about the point stabilizer group and the coset stabilizer group, like the definitions, why they are used in group theory, who developed them and there importance.

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There is no such things as "the point stabilizer group" or "the coset stabilizing group". Given a group $G$, a set $S$, and an action of $G$ on $S$, each point of $S$ defines a point stabilizer in $G$; given any group $G$, each subgroup $H$ of $G$ defines a set on which $G$ acts, namely the cosets of $H$, which in turn yields "point" stabilizers. Since the "points" are cosets, they are often called "coset stabilizer group". So your question is really about group actions. What do you know about group actions? – Arturo Magidin Jun 8 '12 at 4:11
There is when working with coset enumeration. Thank you anyway. – HowardRoark Jun 8 '12 at 4:12
@ArturoMagidin I do not know much about group actions. I would appreciate if you could tell me more about them :) – HowardRoark Jun 8 '12 at 4:15
@HowardRoark: My point is that your use of the singular determinate article "the" suggests that there is only one group (in the entire universe) that is called "the point stabilizer group", and only one group (in the entire universe) that is called "the coset stabilizer group". This is simply not the case. When you are working with coset enumeration, you are working within a specific group and a specific subgroup, and are talking about the stabilizers of that particular action. – Arturo Magidin Jun 8 '12 at 4:19
@HowardRoark: Then you need to include all of that context in the question, to make it clear that it is that particular kind of stabilizer that you are talking about. As for good books that discuss group actions, Rotman's textbook has a good chapter on it, and Neumann, Stoy, and Thompson's Groups and Geometry discusses pretty much all of group theory from the point of view of group actions. – Arturo Magidin Jun 8 '12 at 4:31