Daniel Frohardt

Groups which produce generalized quadrangles

The structure of groups which produce generalized quadrangles under a construction of Kantor is investigated. It is shown in particular that if the quadrangles parameter s and t satisfy s ⩾ t then both are powers of the same prime number.

Geometric hyperplanes in generalized hexagons of order (2,2)

We present a detailed study of the geometric hyperplanes of the two generalized hexagons of order (2,2). This leads to concrete descriptions of th universal embeddings of these hexagons, as well as a description of the G 2(2)-orbits on the Lie algebra g 2(2), illustrating some of the anomalies of this algebra. As a byproduct of our investigations, we develop some general theory that can be applied to other incidence systems with 3 points per line.

Groups with a large number of large disjoint subgroups

It is known that if G is a group of order 4N2, and G contains N mutually disjoint subgroups of order 2N, then the nonidentity elements of these subgroups form a difference set in G. Gluck recently discovered a nonabelian example with N = 4 and showed it to be the only case with N = 4 and G not elementary abelian. We show here that the only examples with N > 4 are elementary abelian 2-groups.

Normality in a Kantor family

Abstract A Kantor family is a collection of subgroups from which a generalized quadrangle can be constructed using Kantors idea. This paper considers the case in which some of the subgroups in the Kantor family or its related family are normal in the ambient finite group G. We show that if two members of a Kantor family are normal in G then G is elementary abelian and that if all members of the related family are normal then G is a p-group.

A trilinear form for the third Janko group

Abstract In this paper we describe the action of the sporadic simple group J3 on a vector space M of dimension 85 over Q(√−3, √−19). In doing so, we shall prove that J3 is uniquely determined, up to isomorphism, by the conditions given by Z. Janko [Ist. Naz. Alta Math. 1 (1968), 25–64] in 1967. We also describe a J3-invariant symmetric trilinear form ϑ on M. This form, which is uniquely determined up to scalar multiplication, can be used to define a multiplication on A = M ⊕ M ∗ that makes A into a non-associative algebra.

Grassmannian Fixed Point Ratios

We provide estimates for the fixed point ratios in the permutation representations of a finite classical group over a field of order q on k-subspaces of its natural n-dimensional module. For sufficiently large n, each element must either have a negligible ratio or act linearly with a large eigenspace. We obtain an asymptotic estimate in the latter case, which in most cases is q−dk where d is the codimension of the large eigenspace. The results here are tailored for our forthcoming proof of a conjecture of Guralnick and Thompson on composition factors of monodromy groups.

Universal embeddings for the 3 D 4 (2) hexagon and J 2 near-octagon

Abstract The universal embeddings overF2 of the generalized hexagon for3D4(2)and the near-octagon forJ2are determined to be the 28-dimensional adjoint modules: for typeD4in the first case, and in the second from the 14-dimensional adjointF4-module forG2(4)containingJ2.

Incidence Matrices, Permutation Characters, and the Minimal Genus of a Permutation Group

Corollary 1.2 Let G be an almost simple classical group with natural module V of dimension n. If G is linear, assume that G does not contain a graph automorphism. Let 2 ≤ k < n − 1, and let K be the stabilizer of a nondegenerate or totally singular k-space of V . Let P be the stabilizer of a singular 1-space of V . Then the permutation module 1P is a submodule of 1 G K unless one of the following holds.

Standard 3-components of type sp(6, 2)

It is shown that if G is a finite simple group with a standard 3-component of type Sp(6, 2) and G satisfies certain 2-local and 3-local conditions then either G is isomorphic to Sp(8, 2) or G is isomorphic to -F4(2).

FIXED POINT RATIOS IN EXCEPTIONAL GROUPS OF RANK AT MOST TWO

ABSTRACT We obtain upper bounds for the fixed point ratios of the faithful primitive representations of those almost simple finite groups G for which is group of exceptional Lie type of Lie rank 1 or 2. These bounds are shown to be either best possible, or, in the case of very close to best possible.