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

From Groupprops

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 with the following property. Let be a set and be a subset of of size . Let be the symmetric group on and be the subgroup of comprising permutations that fix every point of . Then, the double coset index of in is at most . Further, if the cardinality of is at least , the double coset index of in is precisely .

Further, we have:

.

## Related facts

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

- Isotropy of finite subset has finite double coset index in finitary symmetric group
- Isotropy of finite subset has finite double coset index in finitary alternating group

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