# Finitary symmetric group is automorphism-faithful in symmetric group

From Groupprops

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Finitary symmetric group (?)) satisfying a particular subgroup property (namely, Automorphism-faithful subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

## Contents

## Statement

Let be any set. Then, the finitary symmetric group is an automorphism-faithful subgroup in : any nontrivial automorphism of that restricts to an automorphism on restricts to a *nontrivial* automorphism on .

## Related facts

- Finitary symmetric group is centralizer-free in symmetric group
- Finitary symmetric group is characteristic in symmetric group
- Symmetric groups on infinite sets are complete
- Transposition-preserving automorphism of finitary symmetric group is induced by conjugation by a permutation

## Facts used

- Finitary symmetric group is centralizer-free in symmetric group
- Finitary symmetric group is normal in symmetric group
- Normal and centralizer-free implies automorphism-faithful

## Proof

The proof follows directly by piecing together facts (1)-(3).