Searcharxiv⌕ Search

arXiv subjects

Andrea Lucchini

Publications and source records attributed to Andrea Lucchini.

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

A generalization of a question asked by B. H. Neumann

Let $w \in F_2$ be a word and let $m$ and $n$ be two positive integers. We say that a finite group $G$ has the $w_{m,n}$-property if however a set $M$ of $m$ elements and a set $N$ of $n$ elements of the group is chosen, there exist at least one element of $x \in M$ and at least one element of $y \in M$ such that $w(x,y)=1.$ Assume that there exists a constant $γ< 1$ such that whenever $w$ is not an identity in a finite group $X$, then the probability that $w(x_1,x_2)=1$ in $X$ is at most $γ.$ If $m\leq n$ and $G$ satisfies the $w_{m,n}$-property, then either $w$ is an identity in $G$ or $|G|$ is bounded in terms of $γ, m$ and $n$. We apply this result to the 2-Engel word.

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

The $A$-Möbius function of a finite group

The Möbius function of the subgroup lettice of a finite group $G$ has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let $A$ be a subgroup of the automorphism group $\rm{Aut}(G)$ of a finite group $G$ and denote by $\mathcal C_A(G)$ the set of $A$-conjugacy classes of subgroups of $G.$ For $H\leq G$ let $[H]_A~=~\{~H^a ~\mid ~a\in ~A\}$ be the element of $\mathcal C_A(G)$ containing $H.$ We may define an ordering in $\mathcal C_A(G)$ in the following way: $[H]_A\leq [K]_A$ if $H^a\leq K$ for some $a\in A$. We consider the Möbius function $μ_A$ of the corresponding poset and analyse its properties and possible applications.

math.GR↗
arXiv2021

Semiregularity and connectivity of the non-$\mathfrak F$ graph of a finite group

Given a class $\mathfrak F$ of finite groups, we consider the graph $\widetildeΓ_{\mathfrak F}(G)$ whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h\rangle\notin\mathfrak F$. Moreover we denote by $\mathcal{I}_{\mathfrak F}(G)$ the set of the isolated vertices of $\widetildeΓ_{\mathfrak F}(G).$ We address the following question: to what extent the fact that $\mathcal{I}_{\mathfrak F}(G)$ is a subgroup of $H$ for any $H\leq G,$ implies that the graph $Γ_{\mathfrak F}G)$ obtained from $\widetildeΓ_{\mathfrak F}(G)$ by deleting the isolated vertices is a connected graph?

math.GR↗
arXiv2021

On the connectivity of the non-generating graph

Given a 2-generated finite group $G$, the non-generating graph of $G$ has as vertices the elements of $G$ and two vertices are adjacent if and only if they are distinct and do not generate $G$. We consider the graph $Σ(G)$ obtained from the non-generating graph of $G$ by deleting the universal vertices. We prove that if the derived subgroup of $G$ is not nilpotent, then this graph is connected, with diameter at most 5. Moreover we give a complete classification of the finite groups $G$ such that $Σ(G)$ is disconnected.

math.GR↗
arXiv2021

Forbidden subgraphs in generating graphs of finite groups

Let $G$ be a $2$-generated group. The generating graph $Γ(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g_1$ and $g_2$ are adjacent if $G = \langle g_1, g_2 \rangle.$ This graph encodes the combinatorial structure of the distribution of generating pairs across $G.$ In this paper we study some graph theoretic properties of $Γ(G)$, with particular emphasis on those properties that can be formulated in terms of forbidden induced subgraphs. In particular we investigate when the generating graph $Γ(G)$ is a cograph (giving a complete description when $G$ is soluble) and when it is perfect (giving a complete description when $G$ is nilpotent and proving, among the others, that $Γ(S_n)$ and $Γ(A_n)$ are perfect if and only if $n\leq 4$). Finally we prove that for a finite group $G$, the properties that $Γ(G)$ is split, chordal or $C_4$-free are equivalent.

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

Primitive permutation IBIS groups

Let $G$ be a finite permutation group on $Ω$. An ordered sequence of elements of $Ω$, $(ω_1,\dots, ω_t)$, is an irredundant base for $G$ if the pointwise stabilizer $G_{(ω_1,\dots, ω_t)}$ is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same size we say that $G$ is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost simple, of affine-type, or of diagonal type. Moreover we prove that a diagonal-type primitive permutation groups is IBIS if and only if it is isomorphic to $PSL(2,2^f)\times PSL(2,2^f)$ for some $f\geq 2,$ in its diagonal action of degree $2^f(2^{2f}-1).$

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

Maximal intersections in finite groups

For a finite group $G$, we investigate the behaviour of four invariants, $\text{MaxDim}(G),$ $\text{MinDim}(G),$ $\text{MaxInt}(G)$ and $\text{MinInt}(G),$ measuring in some way the width and the height of the lattice $\mathcal M(G)$ consisting of the intersections of the maximal subgroups of $G.$

math.GR↗
arXiv2020

The non-$\mathfrak F$-graph of a finite group

Given a formation $\mathfrak F$, we consider the graph whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h \rangle \notin\mathfrak F$. We are interested in the two following questions. Is the set of the isolated vertices of this graph a subgroup of $G$? Is the subgraph obtained by deleting the isolated vertices a connected graph?

math.GR↗
arXiv2020

The virtually generating graph of a profinite group

We consider the graph $Γ_{\rm{virt}}(G)$ whose vertices are the elements of a finitely generated profinite group $G$ and where two vertices $x$ and $y$ are adjacent if and only if they topologically generate an open subgroup of $G$. We investigate the connectivity of the graph $Δ_{\rm{virt}}(G)$ obtained from $Γ_{\rm{virt}}(G)$ by removing its isolated vertices. In particular we prove that for every positive integer $t$, there exists a finitely generated prosoluble group $G$ with the property that $Δ_{\rm{virt}}(G)$ has precisely $t$ connected components. Moreover we study the graph $\tilde Γ_{\rm{virt}}(G)$, whose vertices are again the elements of $G$ and where two vertices are adjacent if and only if there exists a minimal generating set of $G$ containing them. In this case we prove that the subgraph $\tilde Δ_{\rm{virt}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

math.GR↗
arXiv2020

Graphs encoding the generating properties of a finite group

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $Γ_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if and only if $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$ We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about $G$ are encoded by these graphs.

math.GR↗
arXiv2020

Connectivity of generating graphs of nilpotent groups

Let $G$ be $2$-generated group. The generating graph of $Γ(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial structure of the distribution of generating pairs across $G$. In this paper we study several natural graph theoretic properties related to the connectedness of $Γ(G)$ in the case where $G$ is a finite nilpotent group. For example, we prove that if $G$ is nilpotent, then the graph obtained from $Γ(G)$ by removing its isolated vertices is maximally connected and, if $|G| \geq 3$, also Hamiltonian. We pose several questions.

math.GR↗
arXiv2020

Profinite groups in which the probabilistic zeta function has no negative coefficients

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)=\sum_{n \in \mathbb N} {a_n(G)}/{n^s}$ is associated, where $a_n(G)=\sum_{|G:H|=n}μ(H, G)$ and $μ(H,G)$ denotes the Möbius function of the lattice of open subgroups of $G.$ Its formal inverse $P_G^{-1}(s)$ is the probabilistic zeta function of $G$. When $G$ is prosoluble, every coefficient of $(P_G(s))^{-1}$ is nonnegative. In this paper we discuss the general case and we produce % existence of a non-prosoluble example and We construct a non-prosoluble finitely generated group $G$ with the same property.

math.GR↗
arXiv2020

Applying the Kövári-Sós-Turán theorem to a question in group theory

Let $m\leq n$ be positive integers and $\mathfrak X$ a class of groups which is closed for subgroups, quotient groups and extensions. Suppose that a finite group $G$ satisfies the condition that for every two subsets $M$ and $N$ of cardinalities $m$ and $n,$ respectively, there exist $x \in M$ and $y \in N$ such that $\langle x, y \rangle\in \mathfrak X.$ Then either $G\in \mathfrak X$ or $|G|\leq \left(\frac{180}{53}\right)^m(n-1).$

math.GR↗
arXiv2020

The independence graph of a finite group

Given a finite group $G,$ we denote by $Δ(G)$ the graph whose vertices are the elements $G$ and where two vertices $x$ and $y$ are adjacent if there exists a minimal generating set of $G$ containing $x$ and $y.$ We prove that $Δ(G)$ is connected and classify the groups $G$ for which $Δ(G)$ is a planar graph.

math.GR↗