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Norberto Gavioli

Publications and source records attributed to Norberto Gavioli.

17 recordsLinked to original sources
arXiv2026

Near-Rings and Skew Braces

This paper adapts Rump's correspondence between radical rings and braces to a more general setting, utilizing a suitable class of near-rings. Given a right near-ring $(R,+,\circ)$ with a multiplicative identity, we define a new operation that yields a monoid. Under natural compatibility conditions expressed via a filtration, this monoid becomes a topological group, yielding a topological right skew brace. This generalization recovers radical-ring braces and successfully applies to near-rings of maps under composition. We provide several explicit applications, demonstrating that the Nottingham group, groups of triangular functions, iterated wreath products of arbitrary groups, and groups of IA-automorphisms of free nilpotent groups naturally arise as multiplicative groups of such topological skew braces.

math.GR↗
arXiv2025

Normality conditions in the Sylow $\boldsymbol{p}$-subgroup of $\boldsymbol{\mathrm{Sym}(p^n)}$ and its associated Lie algebra

In this work, we give a description of the structure of the normal subgroups of a Sylow $p$-subgroup $W_n$ of $\mathrm{Sym}(p^n)$, showing that they contain a term from the lower central series with bounded index. To this end, we explicitly determine the terms of the upper and the lower central series of $W_n$. We provide a similar description of these series in the Lie algebra associated to $W_n$, giving a new proof of the equality of their terms in both the group and the algebra contexts. Finally, we calculate the growth of the normalizer chain starting from an elementary abelian regular subgroup of $W_n$.

math.GR↗
arXiv2024

Transfinite hypercentral iterated wreath product of integral domains

Starting with an integral domain $D$ of characteristic $0$, we consider a class of iterated wreath product $W_n$ of $n$ copies of $D$. In order that $W_n$ be transfinite hypercentral, it is necessary to restrict to the case of wreath products defined by way of numerical polynomials. We also associate to each of these groups a Lie ring, providing a correspondence preserving most of the structure. This construction generalizes a result of \cite{netreba} which characterizes the Lie algebras associated to the Sylow \(p\)-subgroups of the symmetric group \(\Sym(p^n)\). As an application, we explore the normalizer chain $\lbrace\mathbf{N}_{i}\rbrace_{i\geq -1}$ starting from the canonical regular abelian subgroup $T$ of $W_n$. Finally, we characterize the regular abelian normal subgroups of $\mathbf{N}_0$ that are isomorphic to $D^n$.

math.GR↗
arXiv2024

A classification of module braces over the ring of $\mathbf{p}$-adic integers

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 classes and the isoclinism classes of such radical algebras are in correspondence with particular equivalence classes of the bilinear forms defined starting from the products of the algebras.

math.GR↗
arXiv2023

Ideally $r$-constrained graded Lie subalgebras of maximal class algebras

Let $E\supseteq F$ 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 the field extension $E\supseteq F$ is finite. A class of ideally $r$-constrained Lie algebras which are not $(r-1)$-constrained is explicitly constructed, for every $r\geq 1$.

math.RA↗
arXiv2023

A modular idealizer chain and unrefinability of partitions with repeated parts

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-2 terms in the chain. Moreover, they proved that the (n-1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(p^n), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over Z_m whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m-1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As an evidence of this, we establish a correspondence between the two constructions in the case m=2.

math.RA↗
arXiv2023

A definitely periodic chain in the integral Lie ring of partitions

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 $\{1,2,\dots, n-1\}$. Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is definitely periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.

math.CO↗
arXiv2022

GGS-groups over primary trees: Branch structures

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 $γ_3(G)$. This is a significant extension of previously known results regarding periodic GGS-groups over primary trees and general GGS-groups in the case $n=1$. We also show that, as in the case $n=1$, a GGS-group generated by a constant vector is not branch.

math.GR↗
arXiv2021

Unrefinable partitions into distinct parts in a normalizer chain

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 replaced with integers whose sum is x obtaining a new partition into distinct parts. We prove here that the (n-1)-th index of the previously mentioned chain is related to the number of unrefinable partitions into distinct parts satisfying a condition on the minimal excludant.

math.CO↗
arXiv2020

Rigid commutators and a normalizer chain

The novel 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 a famous Euler's partition theorem.

math.GR↗
arXiv2020

A Chain of Normalizers in the Sylow $2$-subgroups of the symmetric group on $2^n$ letters

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, for large values of $n$, the index of a normalizer in the consecutive one does not depend on $n$. Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into at least two distinct parts.

math.GR↗
arXiv2020

An Authenticated Key Scheme over Elliptic Curves for Topological Networks

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 this reason, the designers of schemes for sensor network are always concerned with the need of scalable and adaptable solutions. In this work, we present a novel elliptic-curve based solution, derived from the previously released cryptographic protocol TAKS, which addresses this issue. We give a formal description of the scheme, built on a two-dimensional vector space over a prime field and over elliptic curves, where node topology is more relevant than node identity, allowing a dynamic handling of the network and reducing the cost of network updates. We also study some security concerns and their relation to the related discrete logarithm problem over elliptic curves.

cs.CR↗
arXiv2018

Regular subgroups with large intersection

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.

math.GR↗
arXiv2017

Beauville $p$-groups of wild type and groups of maximal class

Let $G$ be a Beauville finite $p$-group. If $G$ exhibits a `good behaviour' with respect to taking powers, then every lift of a Beauville structure of $G/Φ(G)$ is a Beauville structure of $G$. We say that $G$ is a Beauville $p$-group of wild type if this lifting property fails to hold. Our goal in this paper is twofold: firstly, we fully determine the Beauville groups within two large families of $p$-groups of maximal class, namely metabelian groups and groups with a maximal subgroup of class at most $2$; secondly, as a consequence of the previous result, we obtain infinitely many Beauville $p$-groups of wild type.

math.GR↗
arXiv2017

Metabelian thin Beauville $p$-groups

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 $|γ_i(G):γ_{i+1}(G)|\le p^2$ for all $i\geq 1$. In this paper, we determine Beauville structures in metabelian thin $p$-groups.

math.GR↗