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John Bamberg

Publications and source records attributed to John Bamberg.

At least 37 records · Page 2Linked to original sources
arXiv2022

Synchronising primitive groups of diagonal type exist

Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$. Furthermore, we show that such groups are non-spreading for all prime powers $q$.

math.GR↗
arXiv2022

Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry

In this paper we show that if $θ$ is a $T$-design of an association scheme $(Ω, \mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), then $θ$ consists of precisely half of the vertices of $(Ω, \mathcal{R})$ or it is a $T'$-design, where $|T'|>|T|$. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$-ovoids of generalised octagons of order $(s, s^2)$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $(s,s^2)$; (iii) the dual polar spaces $\mathsf{DQ}(2d, q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d \ge 3$; (iv) the Penttila-Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila-Williford scheme in $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.

math.CO↗
arXiv2022

Codes and Designs in Johnson Graphs From Symplectic Actions on Quadratic Forms

The Johnson graph $J(v, k)$ has as vertices the $k$-subsets of $\mathcal{V}=\{1,\ldots, v\}$, and two vertices are joined by an edge if their intersection has size $k-1$. An \emph{$X$-strongly incidence-transitive code} in $J (v, k)$ is a proper vertex subset $Γ$ such that the subgroup $X$ of graph automorphisms leaving $Γ$ invariant is transitive on the set $Γ$ of `codewords', and for each codeword $Δ$, the setwise stabiliser $X_Δ$ is transitive on $Δ\times (\mathcal{V}\setminus Δ)$. We classify the \emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for which $X$ is the symplectic group $\mathrm{Sp}_{2n}(2)$ acting as a $2$-transitive permutation group of degree $2^{2n-1}\pm 2^{n-1}$, where the stabiliser $X_Δ$ of a codeword $Δ$ is contained in a \emph{geometric} maximal subgroup of $X$. In particular, we construct two new infinite families of strongly incidence-transitive codes associated with the reducible maximal subgroups of $\mathrm{Sp}_{2n}(2)$.

math.CO↗
arXiv2021

Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

A $pseudo$-$oval$ of a finite projective space over a finite field of odd order $q$ is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order $(q^n,q^n)$ and a Laguerre plane of order $q^n$ (for some $n$). In setting out a programme to construct new generalised quadrangles, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric ${Q}^-(5,q)$, non-equivalent to the $classical$ $example$, a so-called $pseudo$-$conic$. To date, every known pseudo-oval of lines of ${Q}^-(5,q)$ is projectively equivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovals satisfying the $perspective$ property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in $Q^-(5,q)$ and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. These association schemes are introduced and explored, and we provide a complete theory of how pseudo-ovals of lines of $Q^-(5,q)$ can be analysed from this viewpoint.

math.CO↗
arXiv2021

Partial linear spaces with a rank 3 affine primitive group of automorphisms

A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of $AΓL_1(q)$. Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.

math.GR↗
arXiv2021

Orbits of Sylow subgroups of finite permutation groups

We say that a finite group $G$ acting on a set $Ω$ has Property $(*)_p$ for a prime $p$ if $P_ω$ is a Sylow $p$-subgroup of $G_ω$ for all $ω\inΩ$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values of $p$ for which the alternating group $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property $(*)_p$ for an allowable prime $p$.

math.GR↗
arXiv2020

No almost simple sporadic group acts primitively on the points of a generalised quadrangle

A generalised quadrangle is a point-line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits, and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as automorphism groups. We show that no almost simple sporadic group can act primitively on the points of a finite (thick) generalised quadrangle. We also present two new ideas contributing towards analysing point-primitive groups acting on generalised quadrangles. The first is the outline and implementation of an algorithm for determining whether a given group can act primitively on the points of some generalised quadrangle. The second is the discussion of a conjecture resulting from observations made in the course of this work: any group acting primitively on the points of a generalised quadrangle must either act transitively on lines or have exactly two line-orbits, each containing half of the lines.

math.CO↗
arXiv2020

A classification of finite locally 2-transitive generalized quadrangles

Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $π$ acts $2$-transitively on the points of $π$, then $π$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{PΓL}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.

math.GR↗
arXiv2019

On $p$-groups with automorphism groups related to the Chevalley group $G_2(p)$

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\mathrm{GL}(7,p)) G_2(p)$ on the Frattini quotient $P/Φ(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p = 3$, our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_2(q)$ on the octonion algebra over $\mathbb{F}_q$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_q[G_2(q)]$-module.

math.GR↗
arXiv2019

Permutations with orders coprime to a given integer

Let $m$ be a positive integer and let $ρ(m,n)$ be the proportion of permutations of the symmetric group ${\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $ρ(n,m)n^{1-\frac{ϕ(m)}{m}}\sim κ_m$ where $κ_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geqslant m$, \[C(m) \left(\frac{n}{m}\right)^{\frac{ϕ(m)}{m}-1} \leqslant ρ(n,m) \leqslant \left(\frac{n}{m}\right)^{\frac{ϕ(m)}{m}-1}\] where $ϕ$ is Euler's totient function.

math.CO↗
arXiv2019

On intriguing sets of the Penttila-Williford association scheme

We investigate intriguing sets of an association scheme introduced by Penttila and Williford (2011) that was the basis for their construction of primitive cometric association schemes that are not $P$-polynomial nor the dual of a $P$-polynomial scheme. In particular, we give examples and characterisation results for the four types of intriguing sets that arise in this scheme.

math.CO↗
arXiv2018

The thickness of Schubert cells as incidence structures

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.

math.RT↗
arXiv2018

On regular induced subgraphs of generalized polygons

The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact value of $c(k, g)$ is known only for some small values of $k, g$ and three infinite families where $g \in \{6, 8, 12\}$ and $k - 1$ is a prime power. These infinite families come from the incidence graphs of generalized polygons. Some of the best known upper bounds on $c(k,g)$ for $g \in \{6, 8, 12\}$ have been obtained by constructing small regular induced subgraphs of these cages. In this paper, we first use the Expander Mixing Lemma to give a general lower bound on the size of an induced $k$-regular subgraph of a regular bipartite graph in terms of the second largest eigenvalue of the host graph. We use this bound to show that the known construction of $(k,6)$-graphs using Baer subplanes of the Desarguesian projective plane is the best possible. For generalized quadrangles and hexagons, our bounds are new. In particular, we improve the known lower bound on the size of a $q$-regular induced subgraphs of the classical generalized quadrangle $\mathsf{Q}(4,q)$ and show that the known constructions are asymptotically sharp. For prime powers $q$, we also improve the known upper bounds on $c(q,8)$ and $c(q,12)$ by giving new geometric constructions of $q$-regular induced subgraphs in the symplectic generalized quadrangle $\mathsf{W}(3,q)$ and the split Cayley hexagon $\mathsf{H}(q)$, respectively. Our constructions show that \[c(q,8) \le 2(q^3 - q\sqrt{q} - q)\] for $q$ an even power of a prime, and \[c(q, 12) \le 2(q^5 - 3q^3)\] for all prime powers $q$. For $q \in \{3,4,5\}$ we also give a computer classification of all $q$-regular induced subgraphs of the classical generalized quadrangles of order $q$.

math.CO↗
arXiv2017

Maximal linear groups induced on the Frattini quotient of a $p$-group

Let $p>3$ be a prime. For each maximal subgroup $H\leqslant\mathrm{GL}(d,p)$ with $|H| \geqslant p^{3d+1}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/Φ(G)$ and $|G| \leqslant p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups $G$ that we exhibit have exponent $p$, and of all such groups $G$ with the desired action of $H$ on $G/Φ(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.

math.GR↗
arXiv2017

The binary $q$-analogue of the Fano plane has a trivial automorphism group

A $q$-analogue of a $t$-design is a set $S$ of subspaces (of dimension $k$) of a finite vector space $V$ over a field of order $q$ such that each $t$ subspace is contained in a constant $λ$ number of elements of $S$. The smallest nontrivial feasible parameters occur when $V$ has dimension $7$, $t=2$, $q=2$, and $k=3$; which is the $q$-analogue of a $2$-$(7,3,1)$ design, the Fano plane. The existence of the binary $q$-analogue of the Fano plane has yet to be resolved, and it was shown by Kiermaier et al. (2016) that such a configuration must have an automorphism group of order at most $2$. We show that the binary $q$-analogue of the Fano plane has a trivial automorphism group.

math.CO↗
arXiv2017

Synchronization and separation in the Johnson schemes

Recently Peter Keevash solved asymptotically the existence question for Steiner systems by showing that $S(t,k,n)$ exists whenever the necessary divisibility conditions on the parameters are satisfied and $n$ is sufficiently large in terms of $k$ and $t$. The purpose of this paper is to make a conjecture which if true would be a significant extension of Keevash's theorem, and to give some theoretical and computational evidence for the conjecture. We phrase the conjecture in terms of the notions (which we define here) of synchronization and separation for association schemes. These definitions are based on those for permutation groups which grow out of the theory of synchronization in finite automata. In this theory, two classes of permutation groups (called \emph{synchronizing} and \emph{separating}) lying between primitive and $2$-homogeneous are defined. A big open question is how the permutation group induced by $S_n$ on $k$-subsets of $\{1,\ldots,n\}$ fits in this hierarchy; our conjecture would give a solution to this problem for $n$ large in terms of $k$. We prove the conjecture in the case $k=4$: our result asserts that $S_n$ acting on $4$-sets is separating for $n\ge10$ (it fails to be synchronizing for $n=9$).

math.GR↗
arXiv2017

On $m$-ovoids of regular near polygons

We generalise the work of Segre (1965), Cameron - Goethals - Seidel (1978), and Vanhove (2011) by showing that nontrivial $m$-ovoids of the dual polar spaces $DQ(2d, q)$, $DW(2d-1,q)$ and $DH(2d-1,q^2)$ ($d\ge 3$) are hemisystems. We also provide a more general result that holds for regular near polygons.

math.CO↗