5 - A geometric characterization of the Monster

Summary

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 N_H(E)/C_H(E) ≅ L₅(2). Let E₁ < E₂ < … < E₅ = E be a chain of subgroups of E, where ∣E_i∣ = 2ⁱ, 1 ≤ i ≤ 5. Then the elements of type i in G(F₁) are the subgroups of H which are conjugate to E_i; two elements are incident if one of the subgroups contains the other.

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