(a) Find all translations that are symmetries of $\mathbb{Z}^2$
(b) Find all isometries in $GO_2$ that are symmetries of $\mathbb{Z}^2$
My solutions:
(a) Firstly, we define a translation $T_z(x) = x + z$. Now, we know that all translations are isometries. All we need now is to find translations for which $T_z(\mathbb{Z}^2) = \mathbb{Z}^2$ to make it a symmetry. We see that this is true for all $z \in \mathbb{Z}^2$. Thus, we can conclude that all translations that are symmetries of $\mathbb{Z}^2$ are translations defined as follows: $T_z(x) =$ {$x + z | x,z \in \mathbb{Z}^2 $}.
(b) So, I know that $GO_2 = $ {$x \rightarrow Ux |$ where $U$ is an orthogonal $2 \times 2$ matrix}. Therefore, I make the general claim that all isometries of $GO_2$ that are symmetries of $\mathbb{Z}^2$ are any function $F \in GO_2$ where $F(x) = Ux$ with $U$ as a $2 \times 2$ orthogonal matrix and $x, UX \in \mathbb{Z}^2$
I feel like I am not making arguments that are strong enough in my answers. Since I am still new to isometries and symmetries, I find it a bit challenging to apply more complex topics to provide evidence for my thinking.
If anyone is able to help point me in the right direction, it would be much appreciated.