Find all automorphisms of the additive group \((\mathbb Q,+)\).
Solution
Let \(\varphi\colon(\QQ,+)\to(\QQ,+)\) be an additive homomorphism and set \[ c=\varphi(1). \]
1. For every \(m\in\ZZ\),\[ \varphi(m)=mc. \]
For \(m>0\), additivity gives \[ \varphi(m) = \varphi(\underbrace{1+\cdots+1}_{m\text{ terms}}) = m\varphi(1) = mc. \] For \(m=0\), additivity gives \(\varphi(0)=0\). If \(m<0\), then the positive case gives \(\varphi(-m)=(-m)c\), while additivity gives \(\varphi(-m)=-\varphi(m)\). Hence \(\varphi(m)=mc\) in every case.
2. For every positive integer \(n\),\[ \varphi(1/n)=c/n. \]
By additivity, \[ n\varphi(1/n) = \varphi(n\cdot1/n) = \varphi(1) = c. \] The unique rational number whose \(n\)-fold sum is \(c\) is \(c/n\).
3. For every \(x\in\QQ\),\[ \varphi(x)=cx. \]
Write \(x=m/n\) with \(m\in\ZZ\) and \(n>0\). By additivity and step 2, \[ \varphi(x) = \varphi\left(m\frac1n\right) = m\varphi(1/n) = \frac{mc}{n} = cx. \]
4. The automorphisms of \((\QQ,+)\) are exactly\[ \boxed{ \Aut(\QQ,+) = \left\{ x\longmapsto cx:c\in\QQ^\times \right\} }. \]
If \(\varphi\) is an automorphism, then \(\varphi(1)\neq0\) by injectivity, so step 3 shows that \(\varphi\) has the displayed form with \(c\in\QQ^\times\).
Conversely, for each \(c\in\QQ^\times\), the map \(x\mapsto cx\) is additive and has inverse \(x\mapsto c^{-1}x\). Hence it is an automorphism.
5. Q.E.D.
Step 4 gives all and only the requested automorphisms.