arXiv subjects
Publications and source records attributed to Andrea Lucchini.
Given a finite group $G,$ we denote by $Δ(G)$ the graph whose vertices are the proper subgroups of $G$ and in which two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle.$ We prove that if there exists a finite nilpotent group $X$ with $Δ(G)\cong Δ(X),$ then $G$ is supersoluble.
Assume that $G$ is a finite group and let $a$ and $b$ be non-negative integers. We define an undirected 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 $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$ Our aim is to estimate the genus, the thickness and the crossing number of the graph $Γ_{a,b}(G)$ when $a$ and $b$ are positive integers.
Let $G$ be 2-generated group. The generating graph $Γ(G)$ of $G$ 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 definition can be extended to a 2-generated profinite group $G,$ considering in this case topological generation. We prove that the set $V(G)$ of non-isolated vertices of $Γ(G)$ is closed in $G$ and that, if $G$ is prosoluble, then the graph $Δ(G)$ obtained from $Γ(G)$ by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinite group $G$ with the property that $Δ(G)$ has $2^{\aleph_0}$ connected components. This implies that the so called "swap conjecture" does not hold for finitely generated profinite groups. We also prove that if an element of $V(G)$ has finite degree in the graph $Γ(G),$ then $G$ is finite.
For a finite group $G$ denote by $γ(L(G))$ the genus of the subgroup graph of $G.$ We prove that $γ(L(G))$ tends to infinity as either the rank of $G$ or the number of prime divisors of $|G|$ tends to infinity.
A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The first author recently showed that $C(G)\le β\sqrt{|G|}$ for some absolute constant $β$. In this paper we show that, when $G$ is soluble, then $β$ is at most $5/3$. We also show that this is best possible. Furthermore, we show that, in general, for each $ε>0$ there exists a constant $c_ε$ such that $C(G)\le (1+ε)\sqrt{|G|}+c_ε$.
Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilizers of chains of regular partitions, and the other type arises by taking stabilizers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices, related to the dual Ore's theorem and to a problem of Kenneth Brown.
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.
Given a finite group $G$, the generating graph $Γ(G)$ of $G$ has as vertices the non-identity elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. Let $G$ be a 2-generated finite group. We prove that $Γ(G)$ is planar if and only if $G$ is isomorphic to one of the following groups: $C_2, C_3, C_4, C_5, C_6, C_2 \times C_2, D_3, D_4, Q_8, C_4\times C_2, D_6.$
Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $\sum_{p\in π(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $π(G)$ the set of prime numbers dividing the order of $G$.
We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of degree $n$ has at most $an^{3/2}$ maximal systems of imprimitivity. When $G$ is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of $G$ containing $H$ is at most $|G:H|-1$.
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.
A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ is said to invariably generate $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The authors recently showed that for each $ε>0$, there exists a constant $c_ε$ such that $C(G)\le (1+ε)\sqrt{|G|}+c_ε$. This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each $α>0$ there exists an absolute constant $δ_α$ such that if $G$ is a finite group and $C(G)>α\sqrt{|G|}$, then $G$ has a section $X/Y$ such that $|X/Y|\geq δ_α\sqrt{|G|}$, and $X/Y\cong \mathbb{F}_q\rtimes H$ for some prime power $q$, with $H\le \mathbb{F}_q^{\times}$.
For a group $G,$ let $Γ(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $Γ^*(G)$ be the subgraph of $Γ(G)$ that is induced by all the vertices of $Γ(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $Γ^*(G)$ is connected. Moreover $\mathrm{diam}(Γ^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $\mathrm{diam}(Γ^*(G))=\infty$ otherwise. If $F$ is the free group of rank 2, then $Γ^*(F)$ is connected and we deduce from $\mathrm{diam}(Γ^*(\mathbb{Z}\times \mathbb{Z}))=\infty$ that $\mathrm{diam}(Γ^*(F))=\infty.$
For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.
Let $G$ be a finite group and let $p_1,\dots,p_n$ be distinct primes. If $G$ contains an element of order $p_1\cdots p_n,$ then there is an element in $G$ which is not contained in the Frattini subgroup of $G$ and whose order is divisible by $p_1\cdots p_n.$
Given a finite group $G,$ let $e(G)$ be expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+κ$ with $κ\sim 2.75239495.$ The number $κ$ is explicitly described in terms of the Riemann zeta function and is best possible. If $G$ is a permutation group of degree $n,$ then either $G=S_3$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor+κ^*$ with $κ^* \sim 1.606695.$
Given a finite group $G$, the generating graph $Γ(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $Γ(G)$ when $G$ is an alternating group or a symmetric group. In particular, we determine the vertices of $Γ(G)$ having even degree and show that $Γ(G)$ is Eulerian if and only if $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.
Let $H$ be an open subgroup of a profinite group that can be expressed as intersection of maximal subgroups of $G.$ Given a positive real number $η,$ we say that $H$ is an $η$-intersection if there exists a family of maximal subgroups $M_1,\ldots,M_t$ such that $H=M_1\cap\ldots\cap M_t$ and $|G:M_1|\cdots|G:M_t|\le |G:H|^η$. We investigate the meaning of this property and its influence on the group structure.