# Infinite group has no non-identity finitary automorphism

## Statement

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

## Facts used

1. The set of fixed points under any automorphism $\sigma$ of $G$ is a subgroup of $G$. This subgroup is often termed the centralizer of $\sigma$, and denoted $C_G(\sigma)$.
2. Proper subgroup of infinite group is coinfinite: For an infinite group, the complement of any proper subgroup is infinite.

## Proof

Given: A group $G$, an automorphism $\sigma$ of $G$ that is a finitary permutation.

To prove: $\sigma$ is the identity automorphism.

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