85 research outputs found
Extended F_4-buildings and the Baby Monster
The Baby Monster group B acts naturally on a geometry E(B) with diagram
c.F_4(t) for t=4 and the action of B on E(B) is flag-transitive. It possesses
the following properties:
(a) any two elements of type 1 are incident to at most one common element of
type 2, and
(b) three elements of type 1 are pairwise incident to common elements of type
2 iff they are incident to a common element of type 5.
It is shown that E(B) is the only (non-necessary flag-transitive)
c.F_4(t)-geometry, satisfying t=4, (a) and (b), thus obtaining the first
characterization of B in terms of an incidence geometry, similar in vein to one
known for classical groups acting on buildings. Further, it is shown that E(B)
contains subgeometries E(^2E_6(2)) and E(Fi22) with diagrams c.F_4(2) and
c.F_4(1). The stabilizers of these subgeometries induce on them flag-transitive
actions of ^2E_6(2):2 and Fi22:2, respectively. Three further examples for t=2
with flag-transitive automorphism groups are constructed. A complete list of
possibilities for the isomorphism type of the subgraph induced by the common
neighbours of a pair of vertices at distance 2 in an arbitrary c.F_4(t)
satisfying (a) and (b) is obtained.Comment: to appear in Inventiones Mathematica
A new near octagon and the Suzuki tower
We construct and study a new near octagon of order which has its
full automorphism group isomorphic to the group and which
contains copies of the Hall-Janko near octagon as full subgeometries.
Using this near octagon and its substructures we give geometric constructions
of the -graph and the Suzuki graph, both of which are strongly
regular graphs contained in the Suzuki tower. As a subgeometry of this octagon
we have discovered another new near octagon, whose order is .Comment: 24 pages, revised version with added remarks and reference
ACTION OF THE CYCLIC GROUP Cn ACTING ON THE DIAGONALS OF A REGULAR n-GON
The main objective of this paper is to investigate the action of the cyclic group........
Graphs, permutations and topological groups
Various connections between the theory of permutation groups and the theory
of topological groups are described. These connections are applied in
permutation group theory and in the structure theory of topological groups.
The first draft of these notes was written for lectures at the conference
Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has
been corrected
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