I need some assistance with this proof:
Problem Statement: What is the crystallographic restriction for a discrete group of isometries whose translation group $L$ has the form $\mathbb{Z}a$ with $a≠0$.
So we recently began the new section on isometries and discrete groups, but I am very lost. I can understand the basic concepts but I have no idea where to begin with some of these proofs.
The only theorem in our notes about Crystallographic restriction is:
Theorem: (Crystallographic restriction)
If $L\subset \mathbb{R}^{2}$ is non-trivial and $H\subset O_2$ is a group of symmetries of $L$, then $H=C_n$ or $H=D_n$ for $n=1,2,3,4,6$
I am still confused about what it means for a group of isometries to be discrete as well
A group $G$ of isometries of $\mathbb{R}^2$ is discrete if $\exists \varepsilon > G$ such that
i) $t_a\in G$ is a translation, then by $a\in \mathbb{R}^2\setminus \left\{0\right\}$, $|a|\ge\varepsilon$.
ii) If $g\in G$ is a rotation by angle $\theta\neq 0$ around some point, then $|\theta|\ge\varepsilon$.
Okay,first of all, should that say $\exists \varepsilon > |G|$? I believe my teacher wrote that wrong, because as far as I know, $\varepsilon > G$ doesn't make logical sense?
So basically a group is discrete if the magnitude of the translations and the angle of rotations in the group are larger than the order of the group? (My professor mentioned something about how the translations and rotations can't be arbitrarily small)?
I've been looking through various textbooks and I still cannot seem to make the connections. I have been able to follow everything in this course relatively well until we got to this section!
I see that the discrete group of isometries are all translations by vector having a magnitude of all integer multiples of a non-zero $a$? So clearly $L$ is non-trivial. How would I define $H$?