Infinite group has no non-identity finitary automorphism

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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.