Trivial automorphism group implies trivial or order two

Contents

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group
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Statement

If the automorphism group of a group is the trivial group, then the group is either itself trivial, or cyclic of order two.

Proof

Proof outline

The proof has three steps:

• The group must be Abelian: Otherwise, it would have nontrivial inner automorphisms
• The group must have exponent two: Otherwise, the map sending an element to its inverse, would be a nontrivial automorphism. In particular, it is an elementary Abelian group, or is a vector space
• The group must be one-dimensional as a vector space, otherwise it will have other automorphisms (the general linear group on a vector space of dimension greater than one, is nontrivial)