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Kamilla Rekvényi

Publications and source records attributed to Kamilla Rekvényi.

12 recordsLinked to original sources
arXiv2026

Co-intersection graphs of nonabelian finite simple groups have diameter two

Let $G$ be a finite group. The co-intersection graph $Δ_G ^c$ of $G$ has as its vertices the nontrivial proper subgroups of $G$, with edges joining those pairs of subgroups which intersect trivially. It is clear that every connected component of $Δ_G ^c$ has diameter at most three. In this Note, we show that when $T$ is a nonabelian finite simple group, $Δ_T$ is connected of diameter exactly two. We also make several observations about the extent to which a finite group is determined by its co-intersection graph, and about the class of finite graphs arising as co-intersection graphs.

math.GR↗
arXiv2026

On the generalised Saxl graphs of permutation groups

A base for a finite permutation group $G \le \mathrm{Sym}(Ω)$ is a subset of $Ω$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of base size two, Burness and Giudici introduced the notion of the Saxl graph. This graph has vertex set $Ω$, with edges between elements if they form a base for $G$. We define a generalisation of this graph that encodes useful information about $G$ whenever $b(G) \ge 2$: here, the edges are the pairs of elements of $Ω$ that can be extended to bases of size $b(G)$. In particular, for primitive groups, we investigate the completeness and arc-transitivity of the generalised graph, and the generalisation of Burness and Giudici's Common Neighbour Conjecture on the original Saxl graph.

math.GR↗
arXiv2026

On the orbital diameter of classical groups in standard action

Let $G$ be a primitive permutation group acting on a finite set $X$. The orbital diameter $\mathrm{diam}(X,G)$ is defined to be the supremum of the diameters of the (connected) orbital graphs of $G$ after disregarding the directions of all edges in the graphs. This invariant is studied in the case when $G$ is an almost simple group in a standard action. A lower bound is given for $\mathrm{diam}(X,G)$ and we provide a partial classification of pairs $(X,G)$ for which the orbital diameter is at most $2$.

math.GR↗
arXiv2025

Separating subsets from their images

Let $G$ be a transitive permutation group acting on $Ω$. In this paper, we introduce and study the parameter ${\bf m}(G)$, which denotes the size of the smallest set of points $A$ such that, for every permutation $g\in G$, $A \cap A^g$ is nonempty. In particular, we focus on deriving general bounds for arbitrary transitive groups, and on the asymptotic behaviour of certain families of primitive groups. We also provide a classification of transitive groups with ${\bf m}(G)$ largest possible, namely with ${\bf m}(G)=\lceil (|Ω|+1) / 2 \rceil$.

math.GR↗
arXiv2025

Groups with conjugacy classes of coprime sizes

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $π$-regular elements for some set of primes $π$, then $x^G y^G$ is a $π$-regular conjugacy class in $G$. The latter statement was previously known for $π$-separable groups $G$ and this generalisation permits us to extend several results concerning the common divisor graph on $p$-regular conjugacy classes, for some prime $p$.

math.GR↗
arXiv2025

Prime simplicial complexes of finite groups

The prime simplicial complex $Π(G)$ of a finite group $G$ is composed of all sets of primes $S$ where $G$ has an element of order the product of primes in $S$, with the subsets partially ordered by inclusion. This complex was introduced by Peter Cameron as the generalisation of the well-studied prime (or Gruenberg-Kegel) graphs. In this paper, we establish new results concerning two key properties of $Π(G)$: recognisability and purity. We demonstrate that recognisability by the prime simplicial complex is strictly stronger than recognisability by the prime graph. Notably, we present the first known example of a group that is recognisable by its prime simplicial complex and spectrum, but not by its prime graph, and is not a direct product of two isomorphic simple groups. Furthermore, we classify groups with pure prime simplicial complexes (i.e., all maximal simplices have the same size) across several infinite families of finite simple groups. We also provide a partial classification for non-abelian finite simple groups whose prime simplicial complex has maximal simplices of size at most 2.

math.GR↗
arXiv2024

On the Orbital Diameter of Primitive Affine Groups

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of primitive affine groups with almost quasisimple point stabilizer. As a consequence we obtain a partial classification of primitive affine groups with orbital diameter less than or equal to 2.

math.GR↗
arXiv2024

On the diameter of intersection graphs of finite groups

The intersection graph $Δ_G$ of a finite group $G$ is a simple graph with vertices the non-trivial proper subgroups of $G$, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Csákány and Pollák in 1969. In this paper we answer two long-standing open questions posed by Csákány and Pollák concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.

math.GR↗
arXiv2024

Involutions in finite simple groups as products of conjugates

Let $G$ be a finite non-abelian simple group, $C$ a non-identity conjugacy class of $G$, and $Γ_C$ the Cayley graph of $G$ based on $C \cup C^{-1}$. Our main result shows that in any such graph, there is an involution at bounded distance from the identity.

math.GR↗
arXiv2022

Acyclic orientations and poly-Bernoulli numbers

The main contribution of this paper is a formula for the number of acyclic orientations of a complete bipartite, $K_{n_1,n_2},$ revealing that it is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$ introduced in 1997 by Kaneko. We also give a simple bijective identification of acyclic orientations and lonesum matrices, confirming a 2008 result of Brewbaker, and show that the poly-Bernoulli numbers behave concavely, as well being symmetric. A second goal is to explore the behaviour of more general complete $r$-partite graphs in the space of the number of acyclic orientations. We prove that the number of acyclic orientations of complete bipartite graphs on $n$ vertices is a unimodular concave function maximised by the Turán graph $T(2,n).$ For tripartite graphs, we derive an explicit formula for its number of its acyclic orientations, for the specific case with a single vertex partition. For more complex complete $r$-partite graphs an algorithmic approach is suggested, taking advantage of the relationship with graph colouring through the result of Stanley, namely the value of the graph colouring polynomial at the value -1. An underlying theme is the exploration of the space of the number of acyclic orientations of graphs. We present various theoretical and computational results, with additional, conjectures for both complete $r$-partite and general graphs to guide future research.

math.CO↗
arXiv2021

On the Orbital Diameter of Groups of Diagonal Type

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of simple diagonal type. As a consequence we obtain a classification of simple diagonal groups with orbital diameter less than or equal to 4. As part of this, we classify all finite simple groups with covering number and conjugacy width at most 3. We also prove some general bounds on the covering number and conjugacy width of groups of Lie type.

math.GR↗