A characterization of 3 - local geometry of M ( 24 )

The largest Fischer 3-transposition group M(24) acts ag-transitively on a 3-local incidence geometry G(M(24)) which is a c-extension of the dual polar space associated with the group O 7 (3). The action of the simple commutator subgroup M(24) 0 is still ag-transitive. We show that G(M(24)) is characterized by its diagram under the ag-transitivity assumption. The result implies in particular that G(M(24)) is simply connected. The geometry G(M(24)) appears as a subgeometry in the Buekenhout { Fischer 3-local geometry G(F 1) of the Monster group. The simple connectedness of G(M(24)) has played a crucial role in the characterization of G(F 1), achieved recently. When determining the possible structure of the parabolic subgroups we have used an unpublished pushing-up result by U. Meierfrankenfeld.