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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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    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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