Some geometric aspects of finite abelian groups

Abstract

Let Π be a finite projective plane admitting a large abelian collineation group. It is well known that this situation may be studied by algebraic means (via a representation by suitable types of difference sets), namely using group rings and algebraic number theory and leading to rather strong nonexistence results. What is less well-known is the fact that the abelian group (and sometimes its group ring) can also be used in a much more geometric way; this will be the topic of the present survey. In one direction, abelian collineation groups may be applied for the construction of interesting geometric objects such as unitals, arcs and (hyper-)ovals, (Baer) subplanes, and projective triangles. On the other hand, this approach makes it sometimes possible to provide simple geometric proofs for non-trivial structural restrictions on the given collineation group, avoiding algebraic machinery.

Keywords

• projective plane
• abelian collineation group
• difference set
• relative difference set
• direct product difference set
• neo-difference set
• oval
• hyperoval
• arc
• baer subplane
• projective triangle
• dembowski-piper classification
• semifield plane