Basic coset geometries

TL;DR

This paper characterizes basic pregeometries with no non-degenerate quotients, linking them to primitive group actions, and constructs infinite families of thick geometries with flag-transitive automorphism groups of various O'Nan-Scott types.

Contribution

It provides a new classification of basic geometries based on primitive group actions and constructs infinite examples for each O'Nan-Scott type.

Findings

Characterization of basic geometries via primitive group actions.

Construction of infinite families of thick geometries.

Identification of automorphism groups of various O'Nan-Scott types.

Abstract

In earlier work we gave a characterisation of pregeometries which are `basic' (that is, admit no `non-degenerate' quotients) relative to two different kinds of quotient operations, namely imprimitive quotients and normal quotients. Each basic geometry was shown to involve a faithful group action, which is primitive or quasiprimitive respectively, on the set of elements of each type. For each O'Nan-Scott type of primitive group, we construct a new infinite family of geometries, which are thick and of unbounded rank, and which admit a flag-transitive automorphism group acting faithfully on the set of elements of each type as a primitive group of the given O'Nan-Scott type.