arXiv subjects
Publications and source records attributed to John Bamberg.
We provide a model of the split Cayley hexagon arising from the Hermitian surface $\mathsf{H}(3,q^2)$, thereby yielding a geometric construction of the Dickson group $G_2(q)$ starting with the unitary group $\mathsf{SU}_3(q)$.
A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less obvious ones. We prove that 3/2-transitive groups are either affine or almost simple, and classify the latter. One of the main steps in the proof is an arithmetic result on the subdegrees of groups of Lie type in characteristic $p$: with some explicitly listed exceptions, every primitive action of such a group is either 2-transitive, or has a subdegree divisible by $p$.
In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order $(5^2,5)$ and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order $(s^2,s)$ for $s \le 11$. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle $H(3,q^2)$.
We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.
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 $(q-1,q+1)$ obtained by Payne derivation from the classical symplectic quadrangle $\mathsf{W}(3,q)$. For $q=p^f$ with $f\geq 2$ we obtain at least two nonisomorphic groups when $p\geq 5$ and at least three nonisomorphic groups when $p=2$ or $3$. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial $p$-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.
We give a new construction of the Mathon perp-system of the five dimensional projective space over the field with three elements, starting from four lines.
We prove that every flock generalised quadrangle contains a hemisystem, and we provide a construction method which unifies our results with the examples of Cossidente and Penttila in the classical case.
The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in particular, the well-studied objects known as \textit{generalised quadrangles} are each partial quadrangles. An \textit{intriguing set} of a generalised quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalised quadrangles by Bamberg, Law and Penttila to partial quadrangles, which surprisingly gives insight into the structure of hemisystems and other intriguing sets of generalised quadrangles.
We show that an elation generalised quadrangle which has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle).