I'm reading the classic book "Geometric algebra" by Emil Artin and I have a question/reference request about a remark in it.
In order to make this question self-contained, let me start by defining some notions. "Affine plane" is meant in the sense of incidence geometry. That is, a configuration of points and lines such that: 1) any two distinct points lie in a unique line, 2) given a line $l$ and a point $P$ not in $l$ there is a unique line parallel to $l$ through $P$, and 3) there are three points which are not collinear. Similarly, by "translation" one means a mapping of the affine plane onto itself that either is the identity, or sends each line to a parallel line and has no fixed points (see definitions 2.3 and 2.5 of the book). We also define the "direction" of a translation $\tau$ as the set of all (parallel) lines $l$ with the property that if $P$ is in $l$ also $\tau(P)$ is in $l$.
It is then proven that for any affine plane the set of all translations form a group with composition, and that if there are two translations with different directions this group is abelian.
Just after it, there is a remark (at the very end of section II.2, beginning of p.58 in my edition) where it is said:
It is conceivable that the geometry contains only translations with a single direction $\pi$ (aside from the identity). It is an undecided question whether $T$ (the group of translations) has to be commutative in this case. Probably there do exist counter-examples, but none is known.
Since this book is more than 50 years old, presumably today more is known about this question.
My questions are:
What are some examples of affine planes where all translations have the same direction?
Is nowadays known if there is some affine plane with nonabelian group of translations (or if on the contrary, the translation group is always abelian)?
I would appreciate any pointers to literature regarding these two questions.
Thanks in advance.