12 research outputs found
Point regular groups of automorphisms of generalised quadrangles
We study the point regular groups of automorphisms of some of the known
generalised quadrangles. In particular we determine all point regular groups of
automorphisms of the thick classical generalised quadrangles. We also construct
point regular groups of automorphisms of the generalised quadrangle of order
obtained by Payne derivation from the classical symplectic
quadrangle . For with we obtain at least two
nonisomorphic groups when and at least three nonisomorphic groups
when or . Our groups include nonabelian 2-groups, groups of exponent 9
and nonspecial -groups. We also enumerate all point regular groups of
automorphisms of some small generalised quadrangles.Comment: some minor changes (including to title) after referee's comment
Singer quadrangles
[no abstract available
Simple groups, product actions, and generalized quadrangles
The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation grouppreserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown thatcannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.</jats:p
Benson\u27s Theorem for Partial Geometries
In 1970 Clark Benson published a theorem in the Journal of Algebra stating a congruence for generalized quadrangles. Since then this theorem has been expanded to other specific geometries. In this thesis the theorem for partial geometries is extended to develop new divisibility conditions for the existence of a partial geometry in Chapter 2. Then in Chapter 3 the theorem is applied to higher dimensional arcs resulting in parameter restrictions on geometries derived from these structures. In Chapter 4 we look at extending previous work with partial geometries with α = 2 to uncover potential partial geometries with higher values of α. Finally the theorem is extended to strongly regular graphs in Chapter 5. In addition we obtain expressions for the multiplicities of the eigenvalues of matrices related to the adjacency matrices of these graphs. Finally, a four lesson high school level enrichment unit is included to provide students at this level with an introduction to partial geometries, strongly regular graphs, and an opportunity to develop proof skills in this new context