Norberto Gavioli

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

Professor

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39
Publications
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152
Citations

Publications

Publications (39)
Article
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...
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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 $\gamma_3(G)$...
Article
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.
Article
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...
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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 .
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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...
Article
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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...
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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...
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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...
Article
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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.
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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...
Preprint
Full-text available
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...
Article
Full-text available
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.
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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...
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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.
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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 $G/\Phi(G)$ 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 determi...
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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.
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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.
Article
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.
Article
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.
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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...
Article
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...
Article
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...
Article
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.
Article
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...
Article
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.
Article
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...
Article
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...
Article
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...
Article
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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...

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Project (1)
Project
The Centre of excellence EX-EMERGE (Centre of EXcellence on Connected, Geo-localized and Cyber-secure vehicles) of the University of L’Aquila was founded in 2019 and it represents a major goal of the broader EMERGE (Light Commercial Vehicles & Emerging Technologies for dual-use in “everyday operations” and “emergency”) research program: that program had been formerly selected and approved by the Inter-ministry Committee for Economic Planning (CIPE) of Italy with act n. 70/2017 in the frame of RESTART, an initiative of the Italian Government that relies on research and technology advances as a driver for the economic and social development of L’Aquila and Abruzzo region after the earthquake of 2009. Executive Committee: Prof. Vittorio Cortellessa, Prof. Alessandro D’Innocenzo, Prof. Gabriele Di Stefano, Prof. Norberto Gavioli, Prof. Costanzo Manes, Prof. Patrizio Pelliccione, Prof. Marco Pratesi, Prof. Fortunato Santucci Director: Prof. Fortunato Santucci http://exemerge.disim.univaq.it/