Norberto Gavioli

Università degli Studi dell'Aquila | Università dell'Aquila · Department of Information Engineering, Computer Science and Mathematics

Starting with an integral domain $D$ of characteristic $0$, we define recursively the wreath product $W_n$ of $n$ copies of $D$. For $W_n$ to be transfinite hypercentral, it is necessary to restrict to the class of wreath products defined by way of numerical polynomials. Building on results of Aragona et al., we explore the normalizer chain $\lbrac...

In this paper we study the $R$-braces $(M,+,\circ)$ such that $M\cdot M$ is cyclic, where $R$ is the ring of $p$-adic and $\cdot$ is the product of the radical $R$-algebra associated to $M$. In particular, we give a classification up to isomorphism in the torsion-free case and up to isoclinism in the torsion case. More precisely, the isomorphism cl...

Let G be a Beauville p-group. If G exhibits a ‘good behaviour’ with respect to taking powers, then every lift of a Beauville structure of G/Φ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-...

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in {1,2,⋯,n-1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \u...

While technological advancements and their deep integration in connected and automated vehicles is a central aspect in the evolving trend of automotive industry, they also depict a growing size attack surface for malicious actors: the latter ones typically aim at exploiting known and unknown security vulnerabilities, with potentially disastrous con...

Let E ⊇ F be a field extension and M a graded Lie algebra of maximal class over E. We investigate the F-subalgebras L of M , generated by elements of degree 1. We provide conditions for L being either ideally r-constrained or not just infinite. We show by an example that those conditions are tight. Furthermore, we determine the structure of L when...

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n...

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n...

For every field \(\mathbb{F}\) which has a quadratic extension \(\mathbb{E}\) we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension 2. We construct such Lie algebras as \(\mathbb{F}\)-subalgebras of Lie algebras M of maximal class over \(\mathbb{E}\)...

We study branch structures in Grigorchuk–Gupta–Sidki groups (GGS-groups) over primary trees, that is, regular rooted trees of degree $$p^n$$ p n for a prime p . Apart from a small set of exceptions for $$p=2$$ p = 2 , we prove that all these groups are weakly regular branch over $$G''$$ G ′ ′ . Furthermore, in most cases they are actually regular b...

We study branch structures in Grigorchuk-Gupta-Sidki groups (GGS-groups) over primary trees, that is, regular rooted trees of degree $p^n$ for a prime $p$. Apart from a small set of exceptions for $p=2$, we prove that all these groups are weakly regular branch over $G"$. Furthermore, in most cases they are actually regular branch over $\gamma_3(G)$...

In the present paper we show that a stem finite p-group G has size bounded by min⁡(p(8d−2log2⁡d+b−4)(b+1)/2,pb(3b+4d−1)/2) where b is the breadth of G and pd is the maximum character degree of G. As a consequence there are only finitely many finite stem p-groups having breadth b and maximum character degree pd.

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow 2-subgroup of \({{\,\mathrm{AGL}\,}}(2,n)\), each term of the chain is defined as the normalizer of the previous one in the symmetric group on \(2^n\) letters. Partial results and computational experiments lead us...

In the present paper we show that a stem finite p-group G has size bounded by min p (8d−2 log 2 d+b−4)(b+1)/2 , p b(3b+4d−1)/2 where b is the breadth of G and p d is the maximum character degree of G. As a consequence there are only finitely many finite stem p-groups having breadth b and maximum character degree p d .

In a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2^n elements it has been show that the growth of the first (n-2) consecutive indices of a certain normalizer chain is linked to the sequence of partitions of integers into distinct parts. Unrefinable partitions into distinct parts are those in which no part x can be r...

Nodes of sensor networks may be resource-constrained devices, often having a limited lifetime, making sensor networks remarkably dynamic environments. Managing a cryptographic protocol on such setups may require a disproportionate effort when it comes to update the secret parameters of new nodes that enter the network in place of dismantled sensors...

For every field $F$ which has a quadratic extension $E$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension $2$. We construct such Lie algebras as $F$-subalgebras of Lie algebras $M$ of maximal class over $E$. We characterise the thin Lie $F$-subal...

The notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on 2^n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to...

In this paper we analyse properties satisfied by certain open normal subgroups in normally constrained pro-p groups and in a spread version of normally constrained pro-p groups. In the case of powerful normally constrained pro-p groups, we exhibit some kind of inheritance properties in certain open normal subgroups.

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow $2$-subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, f...

Nodes of sensor networks may be resource-constrained devices, often having a limited lifetime, making sensor networks remarkably dynamic environments. Managing a cryptographic protocol on such setups may require a disproportionate effort when it comes to update the secret parameters of new nodes that enter the network in place of dismantled sensors...

In this paper, we study the relationships between the elementary abelian regular subgroups and the Sylow 2-subgroups of their normalisers in the symmetric group \({{\,\mathrm{Sym}\,}}({{\,\mathrm{\mathbb {F}}\,}}_2^n)\), in view of the interest that they have recently raised for their applications in symmetric cryptography.

In this paper we analyze a list of general properties of waists and waist pairs in a pro-p group, these being subgroups or pairs of subgroups comparable in some sense with respect to inclusion with any open normal subgroup of the pro-p group. In the case of waist pairs of a pro-p group we get some characterizations of them, and give a way of buildi...

In this paper we study the relationships between the elementary abelian regular subgroups and the Sylow $2$-subgroups of their normalisers in the symmetric group $\mathrm{Sym}(\mathbb{F}_2^n)$, in view of the interest that they have recently raised for their applications in symmetric cryptography.

A non-cyclic finite $p$-group $G$ is said to be thin if every normal subgroup of $G$ lies between two consecutive terms of the lower central series and $|\gamma_i(G):\gamma_{i+1}(G)|\le p^2$ for all $i\geq 1$. In this paper, we determine Beauville structures in metabelian thin $p$-groups.

In this paper we study the existence of at least one non-inner automorphism of order p in a finite normally constrained p-group when p is an odd prime.

In this paper we introduce the notion of finite virtual length for profinite groups (that is, every series has a bounded number of infinite factors) and we prove a Jordan-Hölder type theorem for profinite groups with finite virtual length. More structural results are provided in the pronilpotent and p-adic analytic cases.

A waist W of a pro-p group G is a subgroup which is comparable with any open normal subgroup of G. The position of W with respect to the terms of a central series of G is studied here. If p is odd, with some natural hypotheses we show that W is a term of both the lower and upper central series of G.

In 1996 Poland and Rhemtulla proved that the number ν(G) of conjugacy classes of non-normal subgroups of a non-Hamiltonian nilpotent group G is at least c − 1, where c is the nilpotency class of G. In this paper we consider the map that associates to every conjugacy class of subgroups of a finite p-group the conjugacy class of the normaliser of any...

Motivated by the study of pro-p groups with finite coclass, we consider the class of pro-p groups with few normal subgroups. This is not a well defined class and we offer several different definitions and study the connections between them. Furthermore, we propose a definition of periodicity for pro-p groups, thus, providing a general framework for...

A pro-p-group G is said to be normally constrained (or, equivalently, of obliquity zero) if every open normal subgroup of G is trapped between two consecutive terms of the lower central series of G. In this paper infinite soluble normally constrained pro-p-groups, for an odd prime p, are shown to be 2-generated. A classification of such groups, up...

In this paper we deal with graded Lie algebras L such that there exists a positive integer r such that for every positive integer i and for every homogeneous ideal I⊈Li the inclusion I⊇Li+r−1 holds. The solvable case and the r=1 case receive a special attention.

A complete map for a group G is a permutation ϕ : G → G such that g �→ gϕ(g) is still a permutation of G. A conjecture of M. Hall and L. J. Paige states that every finite group whose Sylow 2-subgroup is non-trivial and non-cyclic admits a com- plete map. In the present paper it is proved that a potential counterexample G of minimal order to this co...

A graded Lie algebra is thin if it is generated by two elements of degree 1 and each of its homogeneous ideals is located between two consecutive terms of the lower central series. In this paper we give a complete classification of the metabelian thin Lie algebras and their graded automorphism groups.

Let K be a field and G be the group of the upper unitriangular (n + 2) ( n + 2) K-matrices with nonzero entries only in the first row and in the last column. Then G has a normal subgroup N with a complement H which are K-vector spaces respectively of dimensions n + 1 and n. In the present paper we show that the orbit of H under a group of automorph...

A bijective mapping $$\emptyset :G \to G $$ defined on a finite group G is complete if the mapping ? defined by $$\eta (x) = x\emptyset (x) $$ , $$x \in G $$ , is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is...

In this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say G and K, must satisfy for the existence of a solvable finite group L containing two isomorphic copies of G and H inducing the same permutation character. Also a construction of L is given as an iterated wreath product, with respect to their...

In this paper there are found necessary and sucient conditions that a pair of solvable nite groups, say G and K, must satisfy for the existence of a solvable nite group L containing two isomorphic copies of G and H inducing the same permutation character. Also a construction of L is given as an iterated wreath product, with respect to their actions...