# Automorphism group of finitary symmetric group equals symmetric group

From Groupprops

## Contents

## Statement

Let be an infinite set. Let be the Symmetric group (?) on , i.e., the group of all permutations on , and be the subgroup of that comprises the finitary permutations. In other words, is the Finitary symmetric group (?) on .

Then, is normal in . Further consider the map sending an element to conjugation by on . This map is an isomorphism.

## Related facts

- Symmetric groups on infinite sets are complete
- Automorphism group of finitary alternating group equals symmetric group

## Facts used

- Finitary symmetric group is normal in symmetric group
- Finitary symmetric group is centralizer-free in symmetric group
- Conjugacy class of transpositions is preserved by automorphisms
- Transposition-preserving automorphism of finitary symmetric group is induced by conjugation by a permutation

## Proof

- By fact (1), the map is well-defined, and by fact (2), is injective.
- By fact (3) every element of sends transpositions to transpositions. Combining this with fact (4), we obtain that every element of comes from an inner automorphism in . Thus, is surjective.