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So I'm reading a paper which assumes the following statement but I would like to be able to prove it.

Let $S=Sym(\mathbb{N})$ denote the symmetric group on the set of natural numbers.

If $\emptyset\subset A \subset \mathbb{N}$ then: $$S_{A}= \{ q \in S : aq\in A,\;\forall a\in A \}$$ is a maximal subgroup of S.

Here is how I would like to prove it. I select $f\in S\setminus S_{{A}}$. I want to show that $\langle S_{{A}}, f \rangle = S$, otherwise we have a contradiction. So i take $g\in S$. If $g\in S_{{A}}$ or $g=f$ we are done so assume $g\in S\setminus (S_{{A}}\cup f )$. How can I show that $g\in \langle S_{{A}}, f \rangle$? I had thought about doing something like finding $h\in\langle S_{{A}}, f \rangle$ such that $gh\in S_{{A}}$ so that $g=ghh^{-1}\in\langle S_{{A}}, f \rangle$ but I can't seem to get it to work. Can anyone help? EDIT: I mean A finite. Why is it enought to show the transposition in the answer is in the group generated by these two?

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    is $A$ finite as in the title of this question? Because then it suffices to show that the permutation $(x\; y)$ with $x \in A$ and $y\in \mathbb N \setminus A$ for fixed $x$ and $y$ is a member of $\langle S_A, f \rangle$2011-03-21
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    I'm not so sure, but consider the action of $S$ on subsets of $\mathbb N$ of order $|A|$. If $|A|>1$ this action is at least 2-transitive, therefore it is primitive, therefore stabilizers are maximal subgroups.2011-03-21
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    If A is infinite, then the setwise-stabilizer of A is not maximal for the same reason that the setwise stabilizer of {1,2} is not maximal in Sym({1,2,3,4}); it is contained in a wreath product subgroup.2011-03-21
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    @Jack Schmidt: you mean if $A$ and $\mathbb N\setminus A$ are both infinite, right? I agree with you then, even though it implies something is wrong with my reasoning above and I'm not sure what.2011-03-21
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    @Myself: yes. If there is a bijection between A and X-A, then the setwise-stabilizer of A in Sym(X) is properly contained in Sym(A) wr Sym(2).2011-03-21
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    EDIT: I mean A finite. Why is it enought to show the transposition in the answer is in the group generated by these two?2011-03-21
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    This question was also asked simultaneously on MathOverflow http://mathoverflow.net/questions/59059/the-setwise-stabiliser-of-a-finite-set-is-maximal-in-symn. Please post a link when you're doing this!2011-03-24

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