Soft planes and groups

TL;DR

This paper explores the construction of finite projective planes through specific group conditions, revealing a class of planes with particular symmetry properties and potential for discovering new planes.

Contribution

It introduces a group-theoretic construction method for finite projective planes characterized by a collineation group fixing a flag and acting transitively on certain flags.

Findings

Includes many known projective planes

Identifies conditions for new planes to exist

Characterizes planes with specific symmetry groups

Abstract

Finite projective planes are constructed using groups that satisfy simple-looking conditions. The resulting projective planes include many known planes and possibly new ones, and are precisely those having a collineation group fixing a flag $(\infty,L_ \infty )$ and transitive on the flags $ (w,W )$ with $w\notin L_ \infty $ and $\infty\notin W$.