On rank 3 residually connected geometries for M_{23}

Abstract

In this paper we determine all rank 3 residually connected geometries for the Mathieu group M 23 for which object stabilizers are maximal subgroups.

On primitive geometries of rank two

“…Our programs gave 81 geometries up to isomorphism. This corrects the number that was obtained in [12], namely 86. Table 7 gives the orbit lengths for every primitive permutation representation of M 23 .…”

“…The problems found in [12] for M 22 are very likely due to an incorrect determination of non-isomorphic pairs of subgroups.…”

“…Primitive coset geometries have been studied since the 1990's, first by a team led by Francis Buekenhout in Brussels [4,5] and later on by Peter Rowley and Nayil Kilic (see for instance [8,9,10,11,12]). These geometries may be used to construct codes, designs, graphs, etc.…”

“…Recently, we had the idea of using rank two primitive geometries to construct new binary codes for the McLaughlin group [13]. More precisely, we examined the binary codes obtained from the row span over F 2 of the adjacency matrices of some strongly regular graphs which occur as subgraphs of the McLaughlin graph, namely those with parameters (105,32,4,12), (120,42,8,18) and (253,112,36,60). These new codes were obtained by computing incidence matrices of rank two geometries whose elementstabilizers are maximal subgroups of the McLaughlin group.…”

Some rank 3 residually connected geometries for M_{23}

“…Rank 2 geometries of M 23 (the Mathieu Group of degree 23) were investigated in [9]. Kilic, in [10], calculated all rank 3 residually connected geometries for the Mathieu group M 23 whose object stabilizer are maximal subgroups. Now we give explicit description these geometries and describe them more clearly using the figures.…”