Infinite group has no non-identity finitary automorphism
- 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.
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 .