Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 2D / ii

Choose an angle $2\pi \alpha$ with $\alpha \notin \mathbb{Q}$. Rotation through that angle has infinite order, so the subgroup it generates is isomorphic to $\mathbb{Z}$.

The group also contains a copy of ${\mathbb{Z}}^2$. Choose real numbers $\alpha ,\beta$ such that $1,\alpha ,\beta$ are linearly independent over $\mathbb{Q}$, for example $\alpha =\sqrt{2}$ and $\beta =\sqrt{3}$. DefineThis is a homomorphism. If $\varphi (m,n)=I$, then $m\alpha +n\beta \in \mathbb{Z}$, and the stated rational independence forces $m=n=0$. Thus $\varphi$ is injective and its image is isomorphic to ${\mathbb{Z}}^2$.