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Alexander A. Ivanov

The sporadic simple group F ₂ known as Fischer's Baby Monster acts flag-transitively on a rank 5 P-geometry G( F ₂ ) \mathcal{G}(F_2 ) . P-geometries are geometries with string diagrams, all of whose nonempty edges except one are projective planes of order 2 and one terminal edge is the geometry of the Petersen graph. Let K \mathcal{K} be a flag-transitive P-geometry of rank 5. Suppose that each proper residue of K \mathcal{K} is isomorphic to the corresponding residue in G( F ₂ ) \mathcal{G}(F_2 ) . We show that in this case K \mathcal{K} is isomorphic to G( F ₂ ) \mathcal{G}(F_2 ) . This result realizes a step in classification of the flag-transitive P-geometries and also plays an important role in the characterization of the Fischer-Griess Monster in terms of its 2-local parabolic geometry.

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