Abstract.Let Τ be the Baby Monster graph which is the graph on the set of {3,4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Τ is locally the commuting graph of central (root) involutions in 2E6(2). The graph Τ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the non-empty intersections of two or more of them form a geometry ℰ(B) with diagram for t=4 and the action of B on ℰ(B) is flag-transitive. We show that ℰ(B) contains subgeometries ℰ(2E6(2)) and ℰ(Fi22) with diagrams c.F4(2) and c.F4(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of 2E6(2):2 and Fi22:2, respectively. The geometries ℰ(B), ℰ(2E6(2)) and ℰ(Fi22) possess 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 if and only if they are incident to a common element of type 5. The paper addresses the classification problem of c.F4(t)-geometries satisfying (a) and (b). We construct three further examples for t=2 with flag-transitive automorphism groups isomorphic to 3⋅2E2:2, E6(2):2 and 226 .F4(2) and one for t=1 with flag-transitive automorphism group 3⋅Fi22:2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F4(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the common neighbours of a pair of vertices at distance 2. Finally, we prove that ℰ(B) is the only c.F4(4)-geometry, satisfying (a) and (b).
[1]
Jacques Tits,et al.
Buildings of Spherical Type and Finite BN-Pairs
,
1974
.
[2]
Alexander A. Ivanov.
Geometry of Sporadic Groups
,
1999
.
[3]
Leonard H. Soicher,et al.
Distance-transitive representations of the sporadic groups
,
1995
.
[4]
A. Ivanov.
A Geometric Characterization of Fischer's Baby Monster
,
1992
.
[5]
Cheryl E. Praeger,et al.
Low rank representations and graphs for sporadic groups
,
1996
.
[6]
Bruce N. Cooperstein,et al.
The 2-spaces of the standard $E_6(q)$-module
,
1988
.
[7]
Satoshi Yoshiara.
On Some Extended Dual Polar Spaces, I
,
1994,
Eur. J. Comb..
[8]
G. James.
The character table of
,
1978
.
[9]
Antonio Pasini,et al.
Locally polar geometries with affine planes
,
1992,
Eur. J. Comb..
[10]
Gary M. Seitz,et al.
On the subgroup structure of exceptional groups of Lie type
,
1998
.
[11]
Dmitrii V. Pasechnik.
Geometric Characterization of the Sporadic Groups Fi22, Fi23, and Fi24
,
1994,
J. Comb. Theory, Ser. A.
[12]
Dmitrii V. Pasechnik.
Extending Polar Spaces of Rank at Least 3
,
1995,
J. Comb. Theory, Ser. A.
[13]
J. Conway,et al.
ATLAS of Finite Groups
,
1985
.
[14]
Bruce N. Cooperstein,et al.
The geometry of root subgroups in exceptional groups. I.
,
1979
.
[15]
Arjeh M. Cohen,et al.
The 2-spaces of the standard E6(q)-module
,
1988
.