Let $G$ be a subgroup of the group of isometries $\operatorname{Isom}\left(\mathbb{R}^{2}\right)$ of the Euclidean plane. What does it mean to say that $G$ is discrete?

Supposing that $G$ is discrete, show that the subgroup $G_{T}$ of $G$ consisting of all translations in $G$ is generated by translations in at most two linearly independent vectors in $\mathbb{R}^{2}$. Show that there is a homomorphism $G \rightarrow O(2)$ with kernel $G_{T}$.

Draw, and briefly explain, pictures which illustrate two different possibilities for $G$ when $G_{T}$ is isomorphic to the additive group $\mathbb{Z}$.