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.