Searcharxiv⌕ Search

arXiv subjects

Pablo Spiga

Publications and source records attributed to Pablo Spiga.

At least 55 records · Page 3Linked to original sources
arXiv2022

A generalization of Szep's conjecture for almost simple groups

We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf N}_G (\langle x\rangle){\bf N}_G(\langle y\rangle)$, where we are denoting by ${\bf N}_G(\langle x\rangle)$ and by ${\bf N}_G(\langle y\rangle)$ the normalizers of the cyclic subgroups $\langle x\rangle$ and $\langle y\rangle$. As a consequence of this result, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf C}_G(x){\bf C}_G(y)$.

math.GR↗
arXiv2022

A classification of finite primitive IBIS groups with alternating socle

Let $G$ be a finite permutation group on $Ω$. An ordered sequence $(ω_1,\ldots,ω_\ell)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups $G$ to the case that the socle of $G$ is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.

math.GR↗
arXiv2022

The Engel graph of almost simple groups

Given a finite group $G$, the Engel graph of $G$ is a directed graph encoding pairs of elements satisfying some Engel word. From the work of Detomi, Lucchini and Nemmi, the strongly connectivity of the Engel graph of an arbitrary group $G$ is reduced to the understanding of the strongly connectivity of the Engel graph of non-abelian simple groups. In this paper, we investigate the strongly connectivity of the Engel graph of finite non-abelian simple groups.

math.GR↗
arXiv2021

Finite transitive groups having many suborbits of cardinality at most two and an application to the enumeration of Cayley graphs

Let $G$ be a finite transitive group on a set $Ω$, let $α\in Ω$ and let $G_α$ be the stabilizer of the point $α$ in $G$. In this paper, we are interested in the proportion $$\frac{|\{ω\in Ω\mid ω\textrm{ lies in a }G_α\textrm{-orbit of cardinality at most two}\}|}{|Ω|},$$ that is, the proportion of elements of $Ω$ lying in a suborbit of cardinality at most two. We show that, if this proportion is greater than $5/6$, then each element of $Ω$ lies in a suborbit of cardinality at most two and hence $G$ is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound $5/6$. We use these results to answer a question concerning the enumeration of Cayley graphs. Given a transitive group $G$ containing a regular subgroup $R$, we determine an upper bound on the number of Cayley graphs on $R$ containing $G$ in their automorphism groups.

math.GR↗
arXiv2021

On $n$-partite digraphical representations of finite groups

A group $G$ admits an \textbf{\em $n$-partite digraphical representation} if there exists a regular $n$-partite digraph $Γ$ such that the automorphism group $\mathrm{Aut}(Γ)$ of $Γ$ satisfies the following properties: $\mathrm{Aut}(Γ)$ is isomorphic to $G$, $\mathrm{Aut}(Γ)$ acts semiregularly on the vertices of $Γ$ and the orbits of $\mathrm{Aut}(Γ)$ on the vertex set of $Γ$ form a partition into $n$ parts giving a structure of $n$-partite digraph to $Γ$. In this paper, for every positive integer $n$, we classify the finite groups admitting an $n$-partite digraphical representation.

math.CO↗
arXiv2021

Cherlin's conjecture on finite primitive binary permutation groups

A permutation group is {\it binary} if its orbits on $k$-tuples, for any integer $k\geq 2$, can be deduced from its orbits on $2$-tuples. Cherlin conjectured that a finite primitive binary permutation group $G$ must lie in one of three known families. In this paper we complete the proof of this conjecture. To do this we study the case where the group $G$ is almost simple of Lie type.

math.GR↗
arXiv2021

On the height and relational complexity of a finite permutation group

Let $G$ be a permutation group on a set $Ω$ of size $t$. We say that $Λ\subseteqΩ$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $Λ$. We define the height of $G$ to be the maximum size of an independent set, and we denote this quantity $\mathrm{H}(G)$. In this paper we study $\mathrm{H}(G)$ for the case when $G$ is primitive. Our main result asserts that either $\mathrm{H}(G)< 9\log t$, or else $G$ is in a particular well-studied family (the "primitive large--base groups"). An immediate corollary of this result is a characterization of primitive permutation groups with large "relational complexity", the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\mathrm{I}(G)$, the maximum length of an irredundant base of $G$, in which case we prove that if $G$ is primitive, then either $\mathrm{I}(G)<7\log t$ or else, again, $G$ is in a particular family (which includes the primitive large--base groups as well as some others).

math.GR↗
arXiv2021

Hypermaps over non-abelian simple groups and strongly symmetric generating sets

A generating pair $x, y$ for a group $G$ is said to be \textbf{\textit{symmetric}} if there exists an automorphism $φ_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{φ_{x,y}}=x^{-1}$ and $y^{φ_{x,y}}=y^{-1}$. Similarly, a group $G$ is said to be \textbf{\textit{strongly symmetric}} if $G$ can be generated with two elements and if all generating pairs of $G$ are symmetric. In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups $G$ such that every orientably regular hypermap with monodromy group $G$ is reflexible.

math.GR↗
arXiv2021

A generalization of Sims conjecture for finite primitive groups and two point stabilizers in primitive groups

In this paper we propose a refinement of Sims conjecture concerning the cardinality of the point stabilizers in finite primitive groups and we make some progress towards this refinement. In this process, when dealing with primitive groups of diagonal type, we construct a finite primitive group $G$ on $Ω$ and two distinct points $α,β\in Ω$ with $G_{αβ}\unlhd G_α$ and $G_{αβ}\ne 1$, where $G_α$ is the stabilizer of $α$ in $G$ and $G_{αβ}$ is the stabilizer of $α$ and $β$ in $G$. In particular, this example gives an answer to a question raised independently by Peter Cameron and by Alexander Fomin.

math.GR↗
arXiv2021

Independent sets of generators of prime power order

A subset $X$ of a finite group $G$ is said to be prime-power-independent if each element in $X$ has prime power order and there is no proper subset $Y$ of $X$ with $\langle Y, Φ(G)\rangle = \langle X, Φ(G)\rangle$, where $Φ(G)$ is the Frattini subgroup of $G$. A group $G$ is $\mathcal{B}_{pp}$ if all prime-power-independent generating sets for $G$ have the same cardinality. We prove that, if $G$ is $\mathcal{B}_{pp}$, then $G$ is solvable. Pivoting on some recent results of Krempa and Stocka, this yields a complete classification of $\mathcal{B}_{pp}$-groups.

math.GR↗
arXiv2020

On fixity of arc-transitive graphs

The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected $2$-arc-transitive graphs of a fixed valence tends to $0$ as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-$L$ graphs, where $L$ is a fixed quasiprimitive graph-restrictive permutation group.

math.CO↗
arXiv2020

Constructing infinitely many half-arc-transitive covers of tetravalent graphs

We prove that, given a finite graph $Σ$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $Σ$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.

math.CO↗
arXiv2020

On triangles in derangement graphs

Given a permutation group $G$, the derangement graph $Γ_G$ of $G$ is the Cayley graph with connection set the set of all derangements of $G$. We prove that, when $G$ is transitive of degree at least $3$, $Γ_G$ contains a triangle. The motivation for this work is the question of how large can be the ratio of the independence number of $Γ_G$ to the size of the stabilizer of a point in $G$. We give examples of transitive groups where this ratio is maximum.

math.CO↗