Symmetric groups on infinite sets are complete
- 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
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.