tailieunhanh - Báo cáo toán học: "Rank three residually connected geometries for M22, revisited"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Rank three residually connected geometries for M22, revisited. | Rank three residually connected geometries for M22 revisited Dimitri Leemans Universite Libre de Bruxelles Departement de Mathematiques Service de Geometrie - CP 216 Boulevard du Triomphe B-1050 Bruxelles Belgium dleemans@ Peter Rowley School of Mathematics University of Manchester Oxford Road Manchester M13 6PL UK Submitted Aug 20 2009 Accepted Dec 22 2009 Published Jan 5 2010 Mathematics Subject Classifications 20D08 51E10 05C25 Abstract The rank 3 residually connected flag transitive geometries r for M22 for which the stabilizer of each object in r is a maximal subgroup of M22 are determined. As a result this deals with the infelicities in Theorem 3 of Kilic and Rowley On rank 2 and rank 3 residually connected geometries for M22. Note di Matematica 22 2003 107-154. 1 Introduction Here we report on calculations carried out using Magma 2 on certain rank 3 geometries for M22 the Mathieu group of degree 22. Putting G M22 the main conclusion is as follows. Theorem 1. Up to conjugacy in Aut G there are 431 rank 3 residually connected hag transitive geometries r satisfying the condition that StaỜG x is a maximal subgroup of G for all x G r. These 431 geometries are tabulated in Section 2 where they are described in terms of the action of M22 on a 22-element set. These geometries may also be downloaded from 6 . This list supersedes that given in Theorem 3 of 5 which not only omits some of the geometries but also contains geometries that should not be there usually because they fail to be flag transitive . We next introduce some notation mostly for use in describing the geometries in Section 2. Our notation for geometries is standard as may be found in THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N4 1 1 . So a geometry r consists of a triple r I where r is a set I the set of types and a symmetric incidence relation on r for which i r u r with each r a non-empty subset of r and iel ii if x G ri y G Fj i j G I and x y then i j. .