Queen Mary University
We discuss some recent work in which the O'Nan--Scott Theorem is used in an attempt to classify certain finite point--line geometries that admit automorphism groups acting primitively on either points or lines. The specific finite geometries considered are the so-called generalised polygons, which include projective planes and several examples constructed from finite simple groups of Lie type. It is conjectured that all of the point- or line-primitive generalised polygons are known, but a proof has eluded many people for decades, despite substantial progress. We will see how the O'Nan--Scott Theorem helps to narrow the search for potential new examples, and, time permitting, discuss how the remaining difficulties raise new group-theoretic questions. Much of the work discussed is either joint with, or inspired by previous work of, several of the other speakers.