arXiv subjects

Martino Garonzi

Publications and source records attributed to Martino Garonzi.

At least 19 recordsLinked to original sources

arXiv2025

On the conjugacy class exponent of the finite simple groups

The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g \in G\}$. We provide upper bounds for $e(G)$ for every finite simple group $G$. In particular, we show that $e(G)\leq 8$ unless $G\in\{\mbox{PSL}_n(q), \mbox{PSU}_n(q), E_6(q),{^2}E_6(q)\}$. For the latter groups $e(G)\leq n,3n+3,36,36$, respectively. In addition, we bound from above the generalized order of semisimple and unipotent elements of finite simple groups of Lie type.

math.GR↗

arXiv2025

Profinite groups with many elements with large nilpotentizer and generalizations

Given a profinite group $G$ and a family $\mathcal{F}$ of finite groups closed under taking subgroups, direct products and quotients, denote by $\mathcal{F}(G)$ the set of elements $g \in G$ such that $\{x \in G\ |\ \langle g,x \rangle \ \mbox{is a pro-}\mathcal{F} \mbox{ group}\}$ has positive Haar measure. We investigate the properties of $\mathcal{F}(G)$ for various choices of $\mathcal{F}$ and its influence on the structure of $G$.

math.GR↗

arXiv2025

On pyramidal groups whose number of involutions is a prime power

A Kirkman Triple System $Γ$ is called $m$-pyramidal if there exists a subgroup $G$ of the automorphism group of $Γ$ that fixes $m$ points and acts regularly on the other points. Such group $G$ admits a unique conjugacy class $C$ of involutions (elements of order $2$) and $|C|=m$. We call groups with this property $m$-pyramidal. We prove that, if $m$ is an odd prime power $p^k$, with $p \neq 7$, then every $m$-pyramidal group is solvable if and only if either $m=9$ or $k$ is odd. The primitive permutation groups play an important role in the proof. We also determine the orders of the $m$-pyramidal groups when $m$ is a prime number.

math.GR↗

arXiv2024

Bounds in terms of the number of cyclic subgroups

A family of groups is called (maximal) cyclic bounded ((M)CB) if, for every natural number $n$, there are only finitely many groups in the family with at most $n$ (maximal) cyclic subgroups. We prove that the family of groups of prime power order is MCB. We also prove that the family of finite groups without cyclic coprime direct factors is CB. As a consequence, a natural number $n \geqslant 10$ is prime if and only if there are only finitely many finite noncyclic groups with precisely $n$ cyclic subgroups.

math.GR↗

arXiv2023

On finite groups with the Magnus Property

We investigate finite groups with the Magnus Property, where a group is said to have the Magnus Property (MP) if whenever two elements have the same normal closure then they are conjugate or inverse conjugate. In particular we observe that a finite MP group is solvable, determine the finite primitive MP groups and determine all the possible orders of the chief factors of a finite MP group. We also determine the MP finite direct products of finite primitive groups, as well as the MP crown-based powers of a finite monolithic primitive group.

math.GR↗

arXiv2023

The structure of $3$-pyramidal groups

A combinatorial block design $D$ is called $3$-pyramidal if there exists a subgroup $G$ of $\mbox{Aut}(D)$ fixing $3$ points and acting regularly on the other points. If this happens, we say that the design is $3$-pyramidal under $G$. In case $D$ is a Kirkman triple system, it is known that such a group $G$ has precisely $3$ involutions, all conjugate to each other. In this paper, we obtain a classification of the groups with this property.

math.GR↗

arXiv2023

On minimal coverings and pairwise generation of some primitive groups of wreath product type

The covering number of a finite group $G$, denoted $σ(G)$, is the smallest positive integer $k$ such that $G$ is a union of $k$ proper subgroups. We calculate $σ(G)$ for a family of primitive groups $G$ with a unique minimal normal subgroup $N$, isomorphic to $A_n^m$ with $n$ divisible by $6$ and $G/N$ cyclic. This is a generalization of a result of E. Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.

math.GR↗

arXiv2022

On the maximal number of elements pairwise generating the finite alternating group

Let $G$ be the alternating group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that, when $n$ varies in the family of composite numbers, $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we explicitly calculate $σ(A_n)$ for $n \geq 21$ congruent to $3$ modulo $18$.

math.GR↗

arXiv2022

On the maximal number of elements pairwise generating the symmetric group of even degree

Let $G$ be the symmetric group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that both functions $σ(G)$ and $ω(G)$ are asymptotically equal to $\frac{1}{2} \binom{n}{n/2}$ when $n$ is even. This, together with a result of S. Blackburn, implies that $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we give a lower bound of $(1-o(1))n$ on $ω(G)$ which is independent of the classification of finite simple groups. We also calculate, for large enough $n$, the clique number of the graph defined as follows: the vertices are the elements of $G$ and two vertices $x,y$ are connected by an edge if $\langle x,y \rangle \geq A_n$.

math.GR↗

arXiv2022

On the diameter of Cayley graphs of classical groups with generating sets containing a transvection

A well-known conjecture of Babai states that if $G$ is any finite simple group and $X$ is a generating set for $G$, then the diameter of the Cayley graph $Cay(G,X)$ is bounded by $\log|G|^c$ for some universal constant $c$. In this paper, we prove such a bound for $Cay(G,X)$ for $G=PSL(n,q),PSp(n,q)$ or $PSU(n,q)$ where $q$ is odd, under the assumptions that $X$ contains a transvection and $q\neq 9$ or $81$.

math.GR↗

arXiv2021

Finite groups, minimal bases and the intersection number

Let $G$ be a finite group and recall that the Frattini subgroup ${\rm Frat}(G)$ is the intersection of all the maximal subgroups of $G$. In this paper, we investigate the intersection number of $G$, denoted $α(G)$, which is the minimal number of maximal subgroups whose intersection coincides with ${\rm Frat}(G)$. In earlier work, we studied $α(G)$ in the special case where $G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that $α(G) \leqslant 4$ for every almost simple group $G$, which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups $G$ we present best possible bounds on a related invariant $β(G)$, which we call the base number of $G$. In this setting, $β(G)$ is the minimal base size of $G$ as we range over all faithful primitive actions of the group and we prove that the bound $β(G) \leqslant 4$ is optimal. Along the way, we study bases for the primitive action of the symmetric group $S_{ab}$ on the set of partitions of $[1,ab]$ into $a$ parts of size $b$, determining the exact base size for $a \geqslant b$. This extends earlier work of Benbenishty, Cohen and Niemeyer.

math.GR↗

arXiv2021

On the Primary Coverings of Finite Solvable and Symmetric Groups

A primary covering of a finite group $G$ is a family of proper subgroups of $G$ whose union contains the set of elements of $G$ having order a prime power. We denote with $σ_0(G)$ the smallest size of a primary covering of $G$, and call it the primary covering number of $G$. We study this number and compare it with its analogous $σ(G)$, the covering number, for the classes of groups $G$ that are solvable and symmetric.

math.GR↗

arXiv2020

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We fill this gap by proving that whenever $v \equiv 39$ (mod 72), or $v \equiv 4^e48 + 3$ (mod $4^e96$) and $e \geq 0$, there exists a KTS on $v$ points having at least $v-3$ automorphisms. This is only one of the consequences of a careful investigation on the KTSs with an automorphism group $G$ acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of $G$, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

math.CO↗

arXiv2020

Alternating groups as products of four conjugacy classes

Let $G$ be the alternating group $\mbox{Alt}(n)$ on $n$ letters. We prove that for any $\varepsilon > 0$ there exists $N = N(\varepsilon) \in \mathbb{N}$ such that whenever $n \geq N$ and $A$, $B$, $C$, $D$ are normal subsets of $G$ each of size at least $|G|^{1/2+\varepsilon}$, then $ABCD = G$.

math.GR↗

arXiv2019

On the minimal dimension of a finite simple group (with an appendix by T.C. Burness and R.M. Guralnick)

Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in $\mathcal{M}$ is not equal to the intersection of any proper subset. The minimal dimension of $G$, denoted ${\rm Mindim}(G)$, is the minimal size of a maximal irredundant set of maximal subgroups of $G$. This invariant was recently introduced by Garonzi and Lucchini and they computed the minimal dimension of the alternating groups. In this paper, we prove that ${\rm Mindim}(G) \leqslant 3$ for all finite simple groups, which is best possible, and we compute the exact value for all non-classical simple groups. We also introduce and study two closely related invariants denoted by $α(G)$ and $β(G)$. Here $α(G)$ (respectively $β(G)$) is the minimal size of a set of maximal subgroups (respectively, conjugate maximal subgroups) of $G$ whose intersection coincides with the Frattini subgroup of $G.$ Evidently, ${\rm Mindim}(G) \leqslant α(G) \leqslant β(G)$. For a simple group $G$ we show that $β(G) \leqslant 4$ and $β(G) - α(G) \leqslant 1$, and both upper bounds are best possible.

math.GR↗

arXiv2019

Maximal irredundant families of minimal size in the alternating group

Let $G$ be a finite group. A family $\mathcal{M}$ of maximal subgroups of $G$ is called `irredundant' if its intersection is not equal to the intersection of any proper subfamily. $\mathcal{M}$ is called `maximal irredundant' if $\mathcal{M}$ is irredundant and it is not properly contained in any other irredundant family. We denote by $\mbox{Mindim}(G)$ the minimal size of a maximal irredundant family of $G$. In this paper we compute $\mbox{Mindim}(G)$ when $G$ is the alternating group on $n$ letters.

math.GR↗

arXiv2019

Group partitions of minimal size

A cover of a finite group $G$ is a family of proper subgroups of $G$ whose union is $G$, and a cover is called minimal if it is a cover of minimal cardinality. A partition of $G$ is a cover such that the intersection of any two of its members is $\{1\}$. In this paper we determine all finite groups that admit a minimal cover that is also a partition. We prove that this happens if and only if $G$ is isomorphic to $C_p \times C_p$ for some prime $p$ or to a Frobenius group with Frobenius kernel being an abelian minimal normal subgroup and Frobenius complement cyclic.

math.GR↗

arXiv2018

On integers that are covering numbers of groups

The covering number of a group $G$, denoted by $σ(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$ or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most $129$ by introducing effective new computational techniques. Furthermore, we prove that, if $\mathscr{F}_1$ is the family of finite groups $G$ such that all proper quotients of $G$ are solvable, then $\mathbb{N}-\{σ(G):G\in \mathscr{F}_1\}$ is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form $(q^m-1)/(q-1)$, where $m\neq3$ and $q$ is a prime power, is a covering number, generalizing a result of Cohn.

math.GR↗