Let $G=\langle a\rangle$ be an infinite cyclic group. $Aut(G)$ be set of all automorphisms of $G$. How to determine $Aut(G)$.
How do we determine set of all automorphisms of an infinte cyclic group?
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group-theory
cyclic-groups
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0Welcome to MSE. Questions posed like this, showing very minimal effort if any, are likely to get closed pretty quickly. To obtain useful responses please specify your attempt and where you got stuck. – 2017-01-07
2 Answers
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Hint 1: A homomorphism $h:G\to H$ is uniquely determined by where it sends the generators of $G$. This is also true for when $H = G$.
Hint 2: Automorphisms must be surjective.
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1Thanks. From this I get that $|Aut(G)| \le 2$. Now I'm trying $|Aut(G)| =2$. – 2017-01-07
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0@user404653 Find two concrete automorphisms, and you're done. – 2017-01-07
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0$f(x)=x$ and $f(x)=x^-1$ – 2017-01-07
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0@user404653 Exactly. – 2017-01-07
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0I showed that $x\neq x^-1$ for all $x$ in $G$. Thus there are exactly two automorphisms. Thank you very much. – 2017-01-07
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Here $G \cong \mathbb{Z}$. You can show that any automorphism $\phi$ can send $1$ either to $1$ or $-1$ (otherwise $\phi$ will not be onto). Hence $Aut(\mathbb{Z}) \cong \mathbb{Z}_{2}$.