Searcharxiv⌕ Search

arXiv subjects

Andrea Lucchini

Publications and source records attributed to Andrea Lucchini.

87 records · Page 5Linked to original sources
arXiv2017

The diameter of the generating graph of a finite soluble group

Let $G$ be a finite 2-generated soluble group and suppose that $\langle a_1,b_1\rangle=\langle a_2,b_2\rangle=G$. If either $G^\prime$ is of odd order or $G^\prime$ is nilpotent, then there exists $b \in G$ with $\langle a_1,b\rangle=\langle a_2,b\rangle=G.$ We construct a soluble 2-generated group $G$ of order $2^{10}\cdot 3^2$ for which the previous result does not hold. However a weaker result is true for every finite soluble group: if $\langle a_1,b_1\rangle=\langle a_2,b_2\rangle=G$, then there exist $c_1, c_2$ such that $\langle a_1, c_1\rangle = \langle c_1, c_2\rangle =\langle c_2, a_2\rangle=G.$

math.GR↗
arXiv2016

Generating sets of finite groups

We investigate the extent to which the exchange relation holds in finite groups $G$. We define a new equivalence relation $\equiv_{\mathrm{m}}$, where two elements are equivalent if each can be substituted for the other in any generating set for $G$. We then refine this to a new sequence $\equiv_{\mathrm{m}}^{(r)}$ of equivalence relations by saying that $x \equiv_{\mathrm{m}}^{(r)}y$ if each can be substituted for the other in any $r$-element generating set. The relations $\equiv_{\mathrm{m}}^{(r)}$ become finer as $r$ increases, and we define a new group invariant $ψ(G)$ to be the value of $r$ at which they stabilise to $\equiv_{\mathrm{m}}$. Remarkably, we are able to prove that if $G$ is soluble then $ψ(G) \in \{d(G), d(G) +1\}$, where $d(G)$ is the minimum number of generators of $G$, and to classify the finite soluble groups $G$ for which $ψ(G) = d(G)$. For insoluble $G$, we show that $d(G) \leq ψ(G) \leq d(G) + 5$. However, we know of no examples of groups $G$ for which $ψ(G) > d(G) + 1$. As an application, we look at the generating graph of $G$, whose vertices are the elements of $G$, the edges being the $2$-element generating sets. Our relation $\equiv_{\mathrm{m}}^{(2)}$ enables us to calculate $\mathrm{Aut}(Γ(G))$ for all soluble groups $G$ of nonzero spread, and give detailed structural information about $\mathrm{Aut}(Γ(G))$ in the insoluble case.

math.GR↗
arXiv2016

The Chebotarev invariant of a finite group: a conjecture of Kowalski and Zywina

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ 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 $G$. Confirming a conjecture of Kowalski and Zywina, we prove that there exists an absolute constant $β$ such that $C(G) \leq β\sqrt{|G|}$ for all finite groups $G.$

math.GR↗
arXiv2015

Maximal subgroups of finite soluble groups in general position

For a finite group $G$ we investigate the difference between the maximum size MaxDim$(G)$ of an "independent" family of maximal subgroups of $G$ and maximum size $m(G)$ of an irredundant sequence of generators of $G$. We prove that MaxDim$(G)=m(G)$ if the derived subgroup of $G$ is nilpotent. However MaxDim$(G)-m(G)$ can be arbitrarily large: for any odd prime $p,$ we construct a finite soluble group with Fitting length 2 satisfying $m(G)=3$ and MaxDim$(G)=p.$

math.GR↗
arXiv2014

Irredundant and minimal covers of finite groups

A cover of a finite non-cyclic group $G$ is a family $\mathcal{H}$ of proper subgroups of $G$ whose union equals $G$. A cover of $G$ is called minimal if it has minimal size, and irredundant if it does not properly contain any other cover. We classify the finite non-cyclic groups all of whose irredundant covers are minimal.

math.GR↗
arXiv2014

Coprime invariable generation and minimal-exponent groups

A finite group $G$ is \emph{coprimely-invariably generated} if there exists a set of generators $\{g_1, ..., g_u\}$ of $G$ with the property that the orders $|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \in G$ the set $\{g_1^{x_1}, ..., g_u^{x_u}\}$ generates $G$. We show that if $G$ is coprimely-invariably generated, then $G$ can be generated with three elements, or two if $G$ is soluble, and that $G$ has zero presentation rank. As a corollary, we show that if $G$ is any finite group such that no proper subgroup has the same exponent as $G$, then $G$ has zero presentation rank. Furthermore, we show that every finite simple group is coprimely-invariably generated. Along the way, we show that for each finite simple group $S$, and for each partition $π_1, ..., π_u$ of the primes dividing $|S|$, the product of the number $k_{π_i}(S)$ of conjugacy classes of $π_i$-elements satisfies $\prod_{i=1}^u k_{π_i}(S) \leq \frac{|S|}{2| Out S|}.$

math.GR↗
arXiv2014

Invariable generation of prosoluble groups

A group $G$ is invariably generated by a subset $S$ of $G$ if $G= s^{g(s)} \mid s\in S$ for each choice of $g(s) \in G$, $s \in S$. Answering two questions posed by Kantor, Lubotzky and Shalev, we prove that the free prosoluble group of rank $d \ge 2$ cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank $d$ and derived length $l$ is invariably generated by precisely $l(d-1)+1$ elements.

math.GR↗
arXiv2014

Invariable generation with elements of coprime prime-power order

A finite group $G$ is coprimely-invariably generated if there exists a set of generators $\{g_1, \ldots, g_d\}$ of $G$ with the property that the orders $|g_1|, \ldots, |g_d|$ are pairwise coprime and that for all $x_1, \ldots, x_d \in G$ the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$. In the particular case when $|g_1|, \ldots, |g_d|$ can be chosen to be prime-powers we say that $G$ is prime-power coprimely-invariably generated. We will discuss these properties, proving also that the second one is stronger than the first, but that in the particular case of finite soluble groups they are equivalent.

math.GR↗
arXiv2013

Rationality of the probabilistic zeta function of finitely generated profinite groups

We discuss whether finiteness properties of a profinite group $G$ can be deduced from the probabilistic zeta function $P_G(s)$. In particular we prove that if $P_G(s)$ is rational and all but finitely many nonabelian composition factors of $G$ are groups of Lie type in a fixed characteristic, then $G$ contains only finitely many maximal subgroups.

math.GR↗
arXiv2013

Covers and Normal Covers of Finite Groups

For a finite non cyclic group $G$, let $γ(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \in \{1,\dots,k\}$ and $g \in G.$ We prove that if $G$ is a noncyclic permutation group of degree $n,$ then $γ(G)\leq (n+2)/2.$ We then investigate the structure of the groups $G$ with $γ(G)=σ(G)$ (where $σ(G)$ is the size of a minimal cover of $G$) and of those with $γ(G)=2.$

math.GR↗
arXiv2013

A finiteness condition on the coefficients of the probabilistic zeta function

We discuss whether finiteness properties of a profinite group $G$ can be deduced from the coefficients of the probabilistic zeta function $P_G(s)$. In particular we prove that if $P_G(s)$ is rational and all but finitely many non abelian composition factors of $G$ are isomorphic to $PSL(2,p)$ for some prime $p$, then $G$ contains only finitely many maximal subgroups.

math.GR↗
arXiv2011

Characterization of finitely generated infinitely iterated wreath products

Given a sequence of $(G_i)_{i \in \N}$ of finite transitive groups of degree $n_i$, let $W_\infty$ be the inverse limit of the iterated permutational wreath products of the first m groups. We prove that $W_\infty$ is (topologically) finitely generated if and only if $\prod_{i=1}^{\infty} (G_i/G_i')$ is finitely generated and the growth of the minimal number of generators of $G_i$ is bounded by $ d...n_1...n_{i-1}$ for a constant $d$. Moreover we give a criterion to decide whether $W_\infty$ is positively finitely generated.

math.GR↗
arXiv2010

Rings as the unions of proper subrings

We describe all possible ways how a ring can be expressed as the union of three of its proper subrings. This is an analogue for rings of a 1926 theorem of Scorza about groups. We then determine the minimal number of proper subrings of the simple matrix ring $M_{n}(q)$ whose union is $M_{n}(q)$.

math.RA↗