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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?2012-06-08
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    There is when working with coset enumeration. Thank you anyway.2012-06-08
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    @ArturoMagidin I do not know much about group actions. I would appreciate if you could tell me more about them :)2012-06-08
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    If you have access to Sciencedirect you can see the paper Enumeration under group action by Harary2012-06-08
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    @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.2012-06-08
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    @HowardRoark: As far as telling you about group actions, that would be entire chapters in groups devoted to group theory. Seems like too broad a question, especially when all I would do is repeat the definitions you can find in any good textbook. I can recommend some books for you to look it up, though.2012-06-08
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    @ArturoMagidin I see what you are saying about "the", I do mean "the" though, if we are doing coset enumeration and we are looking at a double coset, we do want "the" point stabilizer group of that double coset specific coset. And in general, coset stabilizer groups are greater than or equal to point stabilizer groups. I would really appreciate the recommendations. Thank you.2012-06-08
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    @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](http://www.amazon.com/Groups-Geometry-Oxford-Science-Publications/dp/0198534515/ref=sr_1_1?ie=UTF8&qid=1339129822&sr=8-1) discusses pretty much all of group theory from the point of view of group actions.2012-06-08
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    @ArturoMagidin My bad. It is tons of info and I did not want to get too specific and end up getting no help. That is why I made it so general. Thank you man, I really appreciate you input.2012-06-08
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    Well, as Arturo explained at first: coset stabilizers **are** point stabilizer, but for a different group action. For the full monty see the early chapters of Dixon & Mortimer *Permutation Groups* (in Springer GTM series).2012-06-08

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