Book contents
- Frontmatter
- Contents
- Introduction
- Photograph
- 1 Automorphisms of solvable groups, Part I
- 2 Automorphisms of solvable groups, Part II
- 3 A survey of groups with a single defining relation
- 4 Some algorithms for computing with finite permutation groups
- 5 Five lectures on group rings
- 6 Buildings and group amalgamations
- 7 Finite presentability of S-arithmetic groups
- 8 Efficient presentations of GL(2, ℤ) and PGL(2, ℤ)
- 9 The commutator map
- 10 Polynomial functions and representations
- 11 On questions of Brauer and Feit
- 12 The Picard group and the modular group
- 13 Factor groups of the lower central series of free products of finitely generated abelian groups
- 14 Lattice ordered groups - a very biased survey
- 15 Totally orthogonal finite groups
- 16 One-relator products of groups
- 17 The Cavicchioli groups are pairwise non-isomorphic
- 18 Congruence and non-congruence subgroups of the modular group: a survey
- 19 Small cancellation theory with non-homogeneous geometrical conditions and application to certain Artin groups
- 20 The Lie algebra associated to the lower central series of a group
- 21 Algebraically closed locally finite groups
- 22 On power-commutative and commutation transitive groups
- 23 Dimension function for discrete groups
- 24 Coset graphs
- 25 Nilpotent quotient algorithms
- 26 Generators of p-groups
- 27 On the matrix groups associated to the isometries of the hyperbolic plane
- 28 A characteristic subgroup of N-stable groups
- 29 The isomorphism problem for integral group rings of finite nilpotent groups
- 30 Embedding the root group geometry of 2F4(q)
- 31 On generalized Frobenius complements
- 32 Subgroups of finite index in soluble groups: I
- 33 Subgroups of finite index in soluble groups: II
- 34 Some interconnections between group theory and logic
- 35 Groups covered by abelian subgroups
- 36 Embeddings of infinite permutation groups
- 37 Maximal subgroups of sporadic groups
30 - Embedding the root group geometry of 2F4(q)
Published online by Cambridge University Press: 05 March 2012
- Frontmatter
- Contents
- Introduction
- Photograph
- 1 Automorphisms of solvable groups, Part I
- 2 Automorphisms of solvable groups, Part II
- 3 A survey of groups with a single defining relation
- 4 Some algorithms for computing with finite permutation groups
- 5 Five lectures on group rings
- 6 Buildings and group amalgamations
- 7 Finite presentability of S-arithmetic groups
- 8 Efficient presentations of GL(2, ℤ) and PGL(2, ℤ)
- 9 The commutator map
- 10 Polynomial functions and representations
- 11 On questions of Brauer and Feit
- 12 The Picard group and the modular group
- 13 Factor groups of the lower central series of free products of finitely generated abelian groups
- 14 Lattice ordered groups - a very biased survey
- 15 Totally orthogonal finite groups
- 16 One-relator products of groups
- 17 The Cavicchioli groups are pairwise non-isomorphic
- 18 Congruence and non-congruence subgroups of the modular group: a survey
- 19 Small cancellation theory with non-homogeneous geometrical conditions and application to certain Artin groups
- 20 The Lie algebra associated to the lower central series of a group
- 21 Algebraically closed locally finite groups
- 22 On power-commutative and commutation transitive groups
- 23 Dimension function for discrete groups
- 24 Coset graphs
- 25 Nilpotent quotient algorithms
- 26 Generators of p-groups
- 27 On the matrix groups associated to the isometries of the hyperbolic plane
- 28 A characteristic subgroup of N-stable groups
- 29 The isomorphism problem for integral group rings of finite nilpotent groups
- 30 Embedding the root group geometry of 2F4(q)
- 31 On generalized Frobenius complements
- 32 Subgroups of finite index in soluble groups: I
- 33 Subgroups of finite index in soluble groups: II
- 34 Some interconnections between group theory and logic
- 35 Groups covered by abelian subgroups
- 36 Embeddings of infinite permutation groups
- 37 Maximal subgroups of sporadic groups
Summary
In the standard interpretation of the 2-local geometry of a group of Lie type defined over a field of even characteristic it is well-known that the geometry of the finite Ree groups of type F4, is that of a generalized octagon with parameters (q,q2), q the order of the field. Outside of the theory of buildings itself this is perhaps most easily seen within an irreducible Fq-module of minimal dimension where each maximal 2-local P is identified with the subspace centralized by 02(P). It is also well-known that this geometry can be realized as a point-line incidence structure inside the group by taking the centers of the non-abelian root groups as the points and the groups generated by the pairs of such centers in a distinr guished orbital as the lines. (This latter approach, of Cooperstein, was the subject of a considerable body of work on the long root groups of exceptional Lie type, [4].) An internal refinement (without reference to a representation space) has been given in [8], explaining the roles of the various root groups (there are three root lengths to consider in a non reduced system; cf. Tits [11]) and resulting in a configuration built on the octagon which resembles a metasymplectic space. The purpose here is to give a direct description of the fundamental elements of this geometry inside a 26-dimensional Fq-space for these groups. Section 1 gives some terse background and suggests a basis for the module which minimizes calculations.
- Type
- Chapter
- Information
- Proceedings of Groups - St. Andrews 1985 , pp. 300 - 304Publisher: Cambridge University PressPrint publication year: 1987