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Let $G$ be a finite group acting on a finite set $X$. If the action is primitive then the stabilizers are maximal subgroups of $G$ (converse also true). Is there any criteria to get maximal subgroups as stabilizers, with some restriction on the "action"?

For ex. the action of $S_3$ on $X=\{1,2,3\}$ is primitive, and so stabilizers are maximal subgroups of $G$, which are Sylow-2 subgroups of $S_3$. But how can we obtain the maximal subgroup $\langle (123)\rangle$?

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    You want to use `\langle` instead of `<` and `\rangle` instead of `>`.2011-05-10
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    The action of $G$ on the left cosets of a maximal subgroup is primitive, and the subgroup is a stabilizer, so you can always realize any maximal subgroup as a stabilizer of *some* action. Did you mean that, or did you mean getting maximal subgroups as stabilizers of a fixed given action?2011-05-10
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    yes..under what action, we get maximal subgroups as stabilizers? (It looks to be useful to get information about automorphism group of $p$ groups.)2011-05-10
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    If $H$ is a maximal subgroup of $G$, $H$ is a stabilizer of the coset $H$ in the primitive action of $G$ on the left cosets of $H$, the action given by left multiplication. So every maximal subgroup is a stabilizer of a primitive action.2011-05-10

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