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Pablo Spiga

Publications and source records attributed to Pablo Spiga.

At least 37 records · Page 2Linked to original sources
arXiv2024

On the number of generators of groups acting arc-transitively on graphs

Given a finite connected graph $Γ$ and a group $G$ acting transitively on the vertices of $Γ$, we prove that the number of vertices of $Γ$ and the cardinality of $G$ are bounded above by a function depending only on the cardinality of $Γ$ and on the exponent of $G$. We also prove that the number of generators of a group $G$ acting transitively on the arcs of a finite graph $Γ$ cannot be bounded by a function of the valency alone.

math.GR↗
arXiv2024

Asymptotic enumeration of Haar graphical representations

This paper represents a significant leap forward in the problem of enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of Cayley (di)graphs show that almost all finite Cayley (di)graphs have the smallest possible automorphism group. Extending the scope of these results, we enumerate (di)graphs admitting a fixed semiregular group of automorphisms with m orbits. Moreover, we consider the more intricate inquiry of prohibiting arcs within each orbit, where the special case m = 2 is known as the problem of finding Haar graphical representations (HGRs). We significantly advance the understanding of HGRs by proving that the proportion of HGRs among Haar graphs of a finite nonabelian group approaches 1 as the group order grows. As a corollary, we obtain an improved bound on the proportion of DRRs among Cayley digraphs in the solution of Morris and the second author to the Babai-Godsil conjecture.

math.CO↗
arXiv2024

Cliques in derangement graphs for innately transitive groups

Given a permutation group $G$, the derangement graph of $G$ is the Cayley graph with connection set the derangements of $G$. The group $G$ is said to be innately transitive if $G$ has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that, if $G$ is innately transitive of degree $n$ and the derangement graph of $G$ has no clique of size $k$, then $n\le f(k)$. Motivation for this work arises from investigations on Erdős-Ko-Rado type theorems for permutation groups.

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arXiv2024

Groups with elements of order 8 do not have the DCI property

Let $k$ be odd, and $n$ an odd multiple of $3$. We prove that $C_k \rtimes C_8$ and $(C_n \times C_3)\rtimes C_8$ do not have the Directed Cayley Isomorphism (DCI) property. When $k$ is also prime, $C_k \rtimes C_8$ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups $C_p \rtimes C_8$ (where $p$ is an odd prime) are only the second known infinite family of groups that have the CI property but do not have the DCI property. This also shows that no group with an element of order $8$ has the DCI property.

math.CO↗
arXiv2024

Haar graphical representations of finite groups and an application to poset representations

Let $R$ be a group and let $S$ be a subset of $R$. The Haar graph $\mathrm{Haar}(R,S)$ of $R$ with connection set $S$ is the graph having vertex set $R\times\{-1,1\}$, where two distinct vertices $(x,-1)$ and $(y,1)$ are declared to be adjacent if and only if $yx^{-1}\in S$. The name Haar graph was coined by Tomaž Pisanski in one of the first investigations on this class of graphs. For every $g\in R$, the mapping $ρ_g:(x,\varepsilon)\mapsto (xg,\varepsilon)$, $\forall (x,\varepsilon)\in R\times\{-1,1\}$, is an automorphism of $\mathrm{Haar}(R,S)$. In particular, the set $\hat{R}:=\{ρ_g\mid g\in R\}$ is a subgroup of the automorphism group of $\mathrm{Haar}(R,S)$ isomorphic to $R$. In the case that the automorphism group of $\mathrm{Haar}(R,S)$ equals $\hat{R}$, the Haar graph $\mathrm{Haar}(R,S)$ is said to be a Haar graphical representation of the group $R$. Answering a question of Feng, Kovács, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions. Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.

math.CO↗
arXiv2024

On the structure of the character degree graphs having diameter three

The structure of the character degree graphs $Δ(G)$, i.e. the prime graphs on the set $\mathrm{cd}(G)$ of the irreducible character degrees of a finite group $G$, such that $G$ is solvable and $Δ(G)$ has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of Mark Lewis.

math.GR↗
arXiv2024

Groups having minimal covering number 2 of diagonal type

Garonzi and Lucchini~\cite{GL} explored finite groups $G$ possessing a normal $2$-covering, where no proper quotient of $G$ exhibits such a covering. Their investigation offered a comprehensive overview of these groups, delineating that such groups fall into distinct categories: almost simple, affine, product action, or diagonal. In this paper, we focus on the family falling under the diagonal type. Specifically, we present a thorough classification of finite diagonal groups possessing a normal $2$-covering, with the attribute that no proper quotient of $G$ has such a covering.

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arXiv2023

On the diameter of Engel graphs

Given a finite group $G$, the Engel graph of $G$ is a directed graph $Γ(G)$ encoding pairs of elements satisfying some Engel word. Namely, $Γ(G)$ is the directed graph, where the vertices are the non-hypercentral elements of $G$ and where there is an arc from $x$ to $y$ if and only if $[x,_ n y] = 1$ for some $n \in \mathbb{N}$. From previous work, it is known that, except for a few exceptions, $Γ(G)$ is strongly connected. In this paper, we give an absolute upper bound on the diameter of $Γ(G)$, when $Γ(G)$ is strongly connected.

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arXiv2023

On the cardinality of irredundant and minimal bases of finite permutation groups

Given a finite permutation group $G$ with domain $Ω$, we associate two subsets of natural numbers to $G$, namely $\mathcal{I}(G,Ω)$ and $\mathcal{M}(G,Ω)$, which are the sets of cardinalities of all the irredundant and minimal bases of $G$, respectively. We prove that $\mathcal{I}(G)$ is an interval of natural numbers, whereas $\mathcal{M}(G,Ω)$ may not necessarily form an interval. Moreover, for a given subset of natural numbers $X \subseteq \mathbb{N}$, we provide some conditions on $X$ that ensure the existence of both intransitive and transitive groups $G$ such that $\mathcal{I}(G,Ω) = X$ and $\mathcal{M}(G,Ω) = X$.

math.GR↗
arXiv2023

On the maximum number of subgroups of a finite group

Given a finite group $R$, we let $\mathrm{Sub}(R)$ denote the collection of all subgroups of $R$. We show that $|\mathrm{Sub}(R)|< c\cdot |R|^{\frac{\log_2|R|}{4}}$, where $c<7.372$ is an explicit absolute constant. This result is asymptotically best possible. Indeed, as $|R|$ tends to infinity and $R$ is an elementary abelian $2$-group, the ratio $$\frac{|\mathrm{Sub}(R)|}{|R|^{\frac{\log_2|R|}{4}}}$$ tends to $c$.

math.GR↗
arXiv2023

Metric dimension of dual polar graphs

A resolving set for a graph $Γ$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension $μ(Γ)$ is the smallest size of a resolving set for $Γ$. We consider the metric dimension of the dual polar graphs, and show that it is at most the rank over $\mathbb{R}$ of the incidence matrix of the corresponding polar space. We then compute this rank to give an explicit upper bound on the metric dimension of dual polar graphs, as well as the halved dual polar graphs.

math.CO↗
arXiv2023

Almost all Cayley maps are mapical regular representations

Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.

math.CO↗
arXiv2022

Normal $2$-coverings of the finite simple groups and their generalizations

Given a finite group $G$, we say that $G$ has weak normal covering number $γ_w(G)$ if $γ_w(G)$ is the smallest integer with $G$ admitting proper subgroups $H_1,\ldots,H_{γ_w(G)}$ such that each element of $G$ has a conjugate in $H_i$, for some $i\in \{1,\ldots,γ_w(G)\}$, via an element in the automorphism group of $G$. We prove that the weak normal covering number of every non-abelian simple group is at least $2$ and we classify the non-abelian simple groups attaining $2$. As an application, we classify the non-abelian simple groups having normal covering number $2$. We also show that the weak normal covering number of an almost simple group is at least two up to one exception. We determine the weak normal covering number and the normal covering number of the almost simple groups having socle a sporadic simple group. Using similar methods we find the clique number of the invariably generating graph of the almost simple groups having socle a sporadic simple group.

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