Infinite sharply multiply transitive groups

TL;DR

This paper surveys sharply n-transitive groups, highlighting the existence of infinite examples that do not originate from fields, thus challenging previous classifications and extending understanding of such groups.

Contribution

It demonstrates the construction of infinite sharply 2-transitive groups not derived from fields, expanding the known landscape beyond classical affine groups.

Findings

Existence of infinite sharply 2-transitive groups outside field-based constructions

Construction methods for non-field arising sharply n-transitive groups

Survey of general sharply n-transitive groups and their structures

Abstract

The finite sharply $2$-transitive groups were classified by Zassenhaus in the 1930's. They essentially all look like the group of affine linear transformations $x\mapsto ax+b$ for some field (or at least near-field) $K$. However, the question remained open whether the same is true for infinite sharply $2$-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus' results might extend might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in o-minimal it was indeed proved that all examples inside the given class arise in this way as affine groups. However, it recently turned out that the reason for the lack of a general proof was the fact that there are plenty of sharply $2$-transitive groups which do not arise from fields or near-fields! In fact, it is not too hard to construct concrete examples (see below). In this note, we survey general sharply $n$-transitive groups and describe how to construct examples not arising from fields.