# Symmetric groups on infinite sets are complete

From Groupprops

## Statement

Let be an infinite set. The symmetric group on , denoted , is a complete group: it is centerless and every automorphism of it is inner.

## Facts used

- Finitary symmetric group is characteristic in symmetric group
- Automorphism group of finitary symmetric group equals symmetric group
- Finitary symmetric group is automorphism-faithful in symmetric group

## Proof

**Given**: is an infinite set, , is an automorphism of .

**To prove**: is inner.

**Proof**: Let be the subgroup of comprising the finitary permutations.

- By fact (1), restricts to an automorphism, say of .
- By fact (2), the automorphism of arises from some inner automorphism, say , of .
- Consider the ratio . The restriction of this automorphism to is which is the identity map. By fact (3), is the identity map on , so . Thus, is inner.