Isotropy of finite subset has finite double coset index in symmetric group

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This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Subgroup of finite double coset index (?)) in a particular group or type of group (namely, Symmetric group (?)).

Statement

There exists a function f: \mathbb{N} \to \mathbb{N} with the following property. Let S be a set and A be a subset of S of size n. Let G be the symmetric group on S and H be the subgroup of G comprising permutations that fix every point of S. Then, the double coset index of H in G is at most f(n). Further, if the cardinality of S is at least 2n, the double coset index of H in G is precisely f(n).

Further, we have:

(n + 1)! \le f(n) \le (2n)!.

Related facts

Some analogous statements for the finitary symmetric group and the finitary alternating group:

In the special case where the subset has size one, we get a subgroup of double coset index two: