## Question

###### Let $G=\mathbb{Q}^{x}$ be the multiplicative group of nonzero rational numbers. If $\alpha=p / q \in G$, where $p$ and $q$ are relatively prime integers, let $\varphi: G \rightarrow G$ be the map which interchanges the primes 2 and 3 in the prime power factorizations of $p$ and $q$ (so, for example, $\varphi\left(2^{4} 3^{11} 5^{1} 13^{2}\right)=3^{4} 2^{11} 5^{1} 13^{2}, \varphi(3 / 16)=\varphi\left(3 / 2^{4}\right)=2 / 3^{4}=2 / 81$, and $\varphi$ is the identity on all rational numbers with n

Let $G=\mathbb{Q}^{x}$ be the multiplicative group of nonzero rational numbers. If $\alpha=p / q \in G$, where $p$ and $q$ are relatively prime integers, let $\varphi: G \rightarrow G$ be the map which interchanges the primes 2 and 3 in the prime power factorizations of $p$ and $q$ (so, for example, $\varphi\left(2^{4} 3^{11} 5^{1} 13^{2}\right)=3^{4} 2^{11} 5^{1} 13^{2}, \varphi(3 / 16)=\varphi\left(3 / 2^{4}\right)=2 / 3^{4}=2 / 81$, and $\varphi$ is the identity on all rational numbers with numerators and denominators relatively prime to 2 and to 3 ). (a) Prove that $\varphi$ is a group isomorphism. (b) Prove that there are infinitely many isomorphisms of the group $G$ to itself. (c) Prove that none of the isomorphisms above can be extended to an isomorphism of the ring $\mathbb{Q}$ to itself. In fact prove that the identity map is the only ring isomorphism of $\mathbb{Q}$.