We present a characterization of the Monster sporadic simple group in terms of its 2-local parabolic geometry.

Introduction

We consider the largest sporadic simple group which is called the Fischer-Griess Monster or the Friendly Giant and is denoted by F1, M or FG. Here we follow the monster terminology and the notation F1.

Let G(F1) be the minimal 2-local parabolic geometry of F1 constructed in. Then G(F1) has rank 5 and belongs to a string diagram all whose nonempty edges except one are projective planes of order 2 and one terminal edge is the triple cover of the generalized quadrangle of order (2,2), related to the nonsplit extension 3·S6. The geometries having diagrams of this shape are called T-geometries.

The geometry G(F1) can be described as follows. The group H ≅ F1 contains an elementary abelian subgroup E of order 25 such that NH(E)/CH(E) ≅ L5(2). Let E1 < E2 < … < E5 = E be a chain of subgroups of E, where ∣Ei∣ = 2i, 1 ≤ i ≤ 5. Then the elements of type i in G(F1) are the subgroups of H which are conjugate to Ei; two elements are incident if one of the subgroups contains the other.

Let {α1, α2, …, α5} be a maximal flag in G(F1) and Hi be the stabilizer of αi in H. Then Hi are called the maximal parabolic subgroups associated with the action of H on G(F1). Without loss of generality we can assume that αi = Ei (clearly Hi = NH(Ei) in this case), 1 ≤ i ≤ 5. Below we present a diagram of stabilizers where under the node of type i the structure of Hi is indicated.