I have known that the only finite group which has only one automorphism is the cyclic group with order less than $2$.
But what's the infinite situation? Is there any infinite group which also satisfies this condition?
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There's also the group of order $1.$ Every infinite group has a nontrivial automorphism, but I think you may need the axiom of choice to prove this in all cases. How about you sketch the proof you know for finite groups in your question, and then we can think about what parts carry over to infinite groups and what more needs to be done? – bof Aug 30 '18 at 09:17
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Just by looking at the Wikipedia page on "automorphisms", one sees reference to the fact that $\mathbb{Z}$ has a unique non-trivial automorphism (when considered as an infinite additive group). In terms of field automorphisms, $\mathbb{R}$ and $\mathbb{Q}$ have no non-trivial aut.s, while $\mathbb{C}$ has uncountably many (with choice), such as complex conjugation. – Isky Mathews Aug 30 '18 at 09:24
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See the comments on infinite groups at this duplicate. – Dietrich Burde Aug 30 '18 at 09:25
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Since any conjugation is identity, it is an abelian group. So I can use the theorem of the structure of finite abelian groups. Then it is easy to find a non-trivial automorphism for others. – Hugo Aug 30 '18 at 09:26