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

Abstract

The largest Fischer 3-transposition group M(24) acts flag-transitively on a 3-local incidence geometry \(\mathcal{G}\)(M(24)) which is a c-extension of the dual polar space associated with the group O7(3). The action of the simple commutator subgroup M(24)′ is still flag-transitive. We show that \(\mathcal{G}\)(M(24)) is characterized by its diagram under the flag-transitivity assumption. The result implies in particular that \(\mathcal{G}\)(M(24)) is simply connected. The geometry \(\mathcal{G}\)(M(24)) appears as a subgeometry in the Buekenhout-Fischer 3-local geometry \(\mathcal{G}\)(F1) of the Monster group. The simple connectedness of \(\mathcal{G}\)(M(24)) has played a crucial role in the characterization of \(\mathcal{G}\)(F1), which has been achieved recently. When determining the possible structure of the parabolic subgroups we have used an unpublished pushing-up result by U. Meierfrankenfeld.