Let ∆ be a simplicial complex on V an n), where each ai ≥ 2. By utilizing the technique of Macaulay's inverse systems, we can explicitly describe the socle of A in terms of ∆. As a consequence, we determine the simplicial complexes, that we will call levelable, for which there exists a tuple (a1,. .. , an) such that A(∆, a1,. .. , an) is a level algebra.

Suppose that an automorphism group G acts flag-transitively on a finite generalized hexagon or octagon S, and suppose that the action on both the point and line set is primitive. We show that G is an almost simple group of Lie type, that is, the socle of G is a simple Chevalley group.

This note is part of a general programme to classify the automorphism groups of finite linear spaces. There have been a number of contributions to this programme, including two recent surveys [8, 3]. One of the most significant contributions was the classification of flag-transitive linear spaces [2]. Since then, the effort has been to classify the line-transitive examples. These fall naturally...