Show $\Bbb{R}^2/C \cong \Bbb{T} \times \Bbb{R}$

Question

Let $\Bbb{T}$ be the multiplicative subgroup of $\Bbb{C}\setminus\{0\}$ with unit modulus and $C$ the additive subgroup of $\Bbb{R}^2$ generated by a non-zero vector. Show that $\Bbb{R}^2/C \cong \Bbb{T} \times \Bbb{R}$.

My initial thought is to in some way encorporate the first isomorphism theorem, much like the case of $\Bbb{R}/\Bbb{Z} \cong \Bbb{T}$, but I cannot quite figure it out. So another thought is to come up with some chain of isomorphisms in which each constituent isomorphism would be easier to handle. However, I have been told that there is a 'natural' isomorphism which, of course, would be preferable.

Hints and directions are appreciated.

Answer 1

Hint: Let $q: \mathbb{R} \to \mathbb{T}$ be the quotient map which you already know. Now you have the sequence of group homomorphisms: $$C \hookrightarrow \mathbb{R} \hookrightarrow \mathbb{R} \times \mathbb{R} \xrightarrow{q \times \operatorname{id}_{\mathbb{R}}} \mathbb{T} \times \mathbb{R}.$$

Answer 2

Hint: Define a function $\varphi:\mathbb{R}^{2}\rightarrow\mathbb{T}\times\mathbb{R}$ by $\varphi((x_{1},x_{2}))=(e^{ix_{1}},x_{2}) $ for all $(x_{1},x_{2})\in\mathbb{R}^{2}$.

Find the kernel and then use the isomorphism theorem. This shows that $\mathbb{R}^{2}/\left\langle v\right\rangle \cong\mathbb{T}\times\mathbb{R}$ for some nonzero $v$. A variation of $\varphi$ can be given to show that $\mathbb{R}^{2}/\left\langle v\right\rangle \cong\mathbb{T}\times\mathbb{R}$ for any nonzero $v$. .