# Infinite group has no non-identity finitary automorphism

From Groupprops

## Statement

Let be an infinite group and be an automorphism of that is a finitary permutation of : the set of points of that are moved by is finite. Then, is the identity automorphism of .

## Facts used

- The set of fixed points under any automorphism of is a subgroup of . This subgroup is often termed the centralizer of , and denoted .
- Proper subgroup of infinite group is coinfinite: For an infinite group, the complement of any proper subgroup is infinite.

## Proof

**Given**: A group , an automorphism of that is a finitary permutation.

**To prove**: is the identity automorphism.

**Proof**: By fact (1), the set of fixed points under in is a subgroup of . By the definition of finitary permutation, the complement is a finite subset of . But fact (2) tells us that if is proper, then is infinite in size. Thus, we must have , so is the identity automorphism on .