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Andrea Lucchini

Publications and source records attributed to Andrea Lucchini.

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

On the connectivity of the generating and rank graphs of finite groups

The generating graph encodes how generating pairs are spread among the elements of a group. For more than ten years it has been conjectured that this graph is connected for every finite group. In this paper, we give evidence supporting this conjecture: we prove that it holds for all but a finite number of almost simple groups and give a reduction to groups without non-trivial soluble normal subgroups. Let $d(G)$ be the minimal cardinality of a generating set for $G$. When $d(G)\geq3$, the generating graph is empty and the conjecture is trivially true. We consider it in the more general setting of the rank graph, which encodes how pairs of elements belonging to generating sets of minimal cardinality spread among the elements of a group. It carries information even when $d(G)\geq3$ and corresponds to the generating graph when $d(G)=2$. We prove that it is connected whenever $d(G)\geq3$, giving tools and ideas that may be used to address the original conjecture.

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arXiv2024

Generating hypergraphs of finite groups

In a recent paper Cameron, Lakshmanan and Ajith began an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective. Following their suggestions, we consider suitable hypergraphs encoding the generating properties of a finite group. In particular, answering a question asked in their paper, we classified the finite solvable groups whose generating hypergraph is the basis hypergraph of a matroid.

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arXiv2024

Abelian supplements in almost simple groups

Let $G$ be a finite almost simple group with socle $G_0$. In this paper we prove that whenever $G/G_0$ is abelian, then there exists an abelian subgroup $A$ of $G$ such that $G=AG_0$. We propose a few applications of this structural property of almost simple groups.

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arXiv2023

On finite $d$-maximal groups

Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up to isomorphism and only partial results have been proven for larger $d$. In this work, we prove that a $d$-maximal group is supersolvable and we give a characterization of $d$-maximality in terms of so-called maximal $(p,q)$-pairs. Moreover, we classify the maximal $(p,q)$-pairs of small rank obtaining, as a consequence, a full classification of the isomorphism classes of $3$-maximal finite groups.

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arXiv2023

Commuting probability for the Sylow subgroups of a finite group

For subsets $X,Y$ of a finite group $G$, let $Pr(X,Y)$ denote the probability that two random elements $x\in X$ and $y\in Y$ commute. Obviously, a finite group $G$ is nilpotent if and only if $Pr(P,Q)=1$ whenever $P$ and $Q$ are Sylow subgroups of $G$ of coprime orders. Suppose that $G$ is a finite group in which for any distinct primes $p,q\inπ(G)$ there is a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ of $G$ such that $Pr(P,Q) \ge ε$. We show that $F_2(G)$ has $ε$-bounded index in $G$. If $G$ is a finite soluble group in which for any prime $p\inπ(G)$ there is a Sylow $p$-subgroup $P$ and a Hall $p'$-subgroup $H$ such that $Pr(P,H)\ge ε$, then $F(G)$ has $ε$-bounded index in $G$. Moreover, we establish criteria for nilpotency and solubility of $G$ such as: If for any primes $p,q\inπ(G)$ the group $G$ has a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ with $Pr(P,Q)>2/3$, then $G$ is nilpotent. If for any primes $p,q\inπ(G)$ the group $G$ has a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ with $Pr(P,Q)>2/5$, then $G$ is soluble.

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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

Group nilpotency from a graph point of view

Let $Γ_G$ denote a graph associated with a group $G$. A compelling question about finite groups asks whether or not a finite group $H$ must be nilpotent provided $Γ_H$ is isomorphic to $Γ_G$ for a finite nilpotent group $G$. In the present work we analyze the problem for different graphs that one can associate with a finite group, both reporting on existing answers and contributing to new ones.

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arXiv2023

The non-two-primes graph of a finite group

To any finite group $G$, we may associate a graph whose vertices are the elements of $G$ and where two distinct vertices $x$ and $y$ are adjacent if and only if the order of the subgroup $\langle x, y\rangle$ is divisible by at least 3 distinct primes. We prove that the subgraph of this graph induced by the non-isolated vertices is connected and has diameter at most 5.

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arXiv2023

The Minimum Generating Set Problem

Let $G$ be a finite group. In order to determine the smallest cardinality $d(G)$ of a generating set of $G$ and a generating set with this cardinality, one should repeat many times the test whether a subset of $G$ of small cardinality generates $G$. We prove that if a chief series of $G$ is known, then the numbers of these generating tests can be drastically reduced. At most $|G|^{13/5}$ subsets must be tested. This implies that the minimum generating set problem for a finite group $G$ can be solved in polynomial time.

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arXiv2023

Probabilistic properties of profinite groups

Let $\mathfrak C$ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group $G$ and an element $x\in G$, we denote by $P_{\mathfrak{C}}(x,G)$ the probability that $x$ and a randomly chosen element of $G$ generate a pro-${\mathfrak C}$ subgroup. We say that a profinite group $G$ is $\mathfrak C$-positive if $P_{\mathfrak{C}}(x,G)>0$ for all $x \in G.$ %Moreover we say that $G$ is $\mathfrak C$-bounded-positive if there exists a positive constant $η$ such that $P_{\mathfrak{C}}(x,G)>η$ for all $x \in G.$ We establish several equivalent conditions for a profinite group to be $\mathfrak C$-positive when $\mathfrak C$ is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite $\mathfrak C$-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.

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arXiv2023

Finite groups satisfying the independence property

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups $H$ contain an element $s$ such that the maximal subgroups of $H$ containing $s$, but not containing the socle of $H$, are pairwise non-conjugate.

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arXiv2022

IBIS soluble linear groups

Let $G$ be a finite permutation group on $Ω.$ An ordered sequence $(ω_1,\dots, ω_t)$ 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. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order.

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arXiv2022

On the soluble graph of a finite group

Let $G$ be a finite insoluble group with soluble radical $R(G)$. In this paper we investigate the soluble graph of $G$, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in $G \setminus R(G)$, with $x$ adjacent to $y$ if they generate a soluble subgroup of $G$. Our main result states that this graph is always connected and its diameter, denoted $δ_{\mathcal{S}}(G)$, is at most $5$. More precisely, we show that $δ_{\mathcal{S}}(G) \leqslant 3$ if $G$ is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that $δ_{\mathcal{S}}(S_n) = 3$ for all $n \geqslant 6$. We also establish the existence of simple groups with $δ_{\mathcal{S}}(G) \geqslant 4$. For instance, we prove that $δ_{\mathcal{S}}(A_{2p+1}) \geqslant 4$ for every Sophie Germain prime $p \geqslant 5$, which demonstrates that our general upper bound of $5$ is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.

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arXiv2022

Strongly generating elements in finite and profinite groups

Given a finite group $G$ and an element $g\in G$, we may compare the expected number $e(G)$ of elements needed to generate $G$ and the expected number $e(G,g)$ of elements of $G$ needed to generate $G$ together with $g.$ We address the following question: how large can the difference $e(G)-e(G,g)$ be?

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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.

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arXiv2022

Minimal invariable generating sets

A subset $S$ of a group $G$ invariably generates $G$ if, when each element of $S$ is replaced by an arbitrary conjugate, the resulting set generates $G.$ An invariable generating set $X$ of $G$ is called minimal if no proper subset of $X$ invariably generates $G.$ We will address several questions related to the behaviour of minimal invariable generating sets of a finite group.

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arXiv2022

The Engel graph of a finite group

For a finite group $G,$ we investigate the direct graph $Γ(G),$ whose vertices are the non-hypercentral elements of $G$ and where there is an edge $x\mapsto y$ if and only if $[x,_ny]=1$ for some $n \in \mathbb N.$ We prove that $Γ(G)$ is always weakly connected and is strongly connected if $G/Z_{\infty}(G)$ is neither Frobenius nor almost simple.

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