arXiv subjects

Riccardo Aragona

Publications and source records attributed to Riccardo Aragona.

At least 19 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↗

arXiv2026

On Zappa--Szép products of a wreathed $2$-group and a cyclic group

Let $n, m \ge 1$, and let $H = C_{2^n}\wr C_2$ be the wreathed $2$-group and $K = C_{2^m}=\langle z \rangle$ a cyclic group. We classify the Zappa--Szép products $G = HK$ in which the base $B \cong C_{2^n}\times C_{2^n}$ of $H$ is normal in $G$. When the matrix $Z$ of the $z$-action on $B$ commutes with the swap $J$ of the two base generators -- equivalently, $Z$ is symmetric circulant -- we classify these products by an explicit system of seven polynomial congruences in a tuple $(p,q,r,s,c)$. Dropping this hypothesis, we obtain a unified classification of all such products with $B$ normal by five congruences on the entries of $Z$ and the parameters $(r,s,c)$, of which the symmetric case is the specialisation $JZ = ZJ$. Finally, we separately treat the classification for the modulus $M = 2^m = 4$, since in this case the congruences degenerate.

math.GR↗

arXiv2026

On Zappa-Szép products of two semidihedral groups

Let $n, m \ge 4$. We classify the Zappa--Szép products $G = HK$ with $H = \langle x\rangle \rtimes \langle y\rangle \cong \mathrm{SD}_{2^n}$ and $K = \langle z\rangle \rtimes \langle w\rangle \cong \mathrm{SD}_{2^m}$, according to the cores of $\langle x\rangle$ and $\langle z\rangle$ in~$G$. First, when both $\langle x\rangle$ and $\langle z\rangle$ are normal in~$G$, we obtain a complete classification of such exact products by an explicit system of six polynomial congruences. Second, when the cores $\langle x\rangle^G$ and $\langle z\rangle^G$ are arbitrary subgroups of $\langle x\rangle$ and $\langle z\rangle$, under the simplifying assumption $[x, z] = 1$ we obtain an analogous classification by twelve congruences together with two order conditions; this is the semidihedral counterpart of the Hu--Yu classification~\cite{HuYu2025} for dihedral groups. In contrast with the dihedral case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \ne 1$, showing that the parameter space in the semidihedral setting is strictly richer than its dihedral analogue.

math.GR↗

arXiv2026

Class-preserving Coleman automorphisms of finite groups with Wreathed Sylow 2-subgroups

We show that if $G$ is a finite group whose Sylow $2$-subgroups are wreathed, then the intersection $\Outc(G) \cap \OutCol(G)$ has odd order, where $\Outc(G)$ and $\OutCol(G)$ denote the class-preserving and Coleman outer automorphism groups, respectively. This implies that $G$ satisfies the normalizer problem for its integral group ring. Combined with earlier work on the dihedral and semidihedral cases, this settles the question for all three families of $2$-groups of $2$-rank two classified by Gorenstein--Walter and Alperin--Brauer--Gorenstein.

math.GR↗

arXiv2026

Class-preserving Coleman Automorphisms of Finite Groups with Semidihedral Sylow 2-Subgroups

In this paper, we prove that finite groups with semidihedral Sylow 2-subgroup have Class-preserving Coleman outer automorphism group of odd order. As a consequence, these groups satisfy the normalizer problem. In particular, we extend some existing results in the literature concerning class-preserving Coleman automorphisms of finite groups with semidihedral Sylow 2-subgroups.

math.GR↗

arXiv2026

A Geometric Characterization of Maximal Unrefinable Partitions via the Keith-Nath Transformation and Young Diagrams

We investigate the combinatorial structure of unrefinable partitions through their correspondence with numerical sets and Young diagrams. Building on the bijection introduced by Keith and Nath, we apply a general geometric criterion that links the unrefinability of a partition directly to the hook lengths of its associated Young diagram. This criterion provides a structural method for the characterization of any unrefinable partition. Using this general framework, we revisit the correspondence results between maximal unrefinable partitions and partitions into distinct parts, previously established using enumerative methods. We provide alternative and purely combinatorial proofs of these bijections, focusing on the rigid symmetry structures of the Young diagrams. In the triangular weight case, we show that the corresponding diagrams are quasi-symmetric, i.e. symmetric up to a single extra column. We extend this analysis to the nontriangular case, showing that the diagrams either exhibit this same quasi-symmetric structure or are perfectly self-conjugate, depending on the maximal part.

math.CO↗

arXiv2025

The number of maximal unrefinable partitions

This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer coincides with the number of suitable partitions into distinct parts, depending on the distance from the successive triangular number.

math.CO↗

arXiv2025

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↗

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

A classification of $\mathbb{F}_{p^k}$-braces using bilinear forms

Let $\mathbb{F}_{p^k}$ be a finite field of odd characteristic $p$. In this paper we give a classification, up to isomorphism, of the associative commutative $\mathbb{F}_{p^k}$-algebras, starting from the connection with their bi-brace structure. Such classification is the generalization in odd characteristic of the result proved by Civino at al. in characteristic $2$.

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

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↗

arXiv2023

Verification and generation of unrefinable partitions

Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, being an unrefinable partition means that none of the parts can be written as the sum of smaller integers without introducing a repetition. We address the algorithmic aspects of unrefinable partitions, such as testing whether a given partition is unrefinable or not and enumerating all the partitions whose sum is a given integer. We design two algorithms to solve the two mentioned problems and we discuss their complexity.

math.CO↗

arXiv2022

On the maximal part in unrefinable partitions of triangular numbers

A partition into distinct parts is refinable if one of its parts $a$ can be replaced by two different integers which do not belong to the partition and whose sum is $a$, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distributions of the parts. We prove a $O(n^{1/2})$-upper bound for the largest part in an unrefinable partition of $n$, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions for triangular numbers, proving that if $n$ is even there exists only one maximal unrefinable partition of $n(n+1)/2$, and that if $n$ is odd the number of such partitions equals the number of partitions of $\lceil n/2\rceil$ into distinct parts. In the second case, an explicit bijection is provided.

math.CO↗

arXiv2022

On the primitivity of the AES-128 key-schedule

The key-scheduling algorithm in the AES is the component responsible for selecting from the master key the sequence of round keys to be xor-ed to the partially encrypted state at each iteration. We consider here the group $Γ$ generated by the action of the AES-128 key-scheduling operation, and we prove that the smallest group containing $Γ$ and all the translations of the message space is primitive. As a consequence, we obtain that no proper and non-trivial subspace can be invariant under its action.

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

On the primitivity of Lai-Massey schemes

In symmetric cryptography, the round functions used as building blocks for iterated block ciphers are often obtained as the composition of different layers providing confusion and diffusion. The study of the conditions on such layers which make the group generated by the round functions of a block cipher a primitive group has been addressed in the past years, both in the case of Substitution Permutation Networks and Feistel Networks, giving to block cipher designers the receipt to avoid the imprimitivity attack. In this paper a similar study is proposed on the subject of the Lai-Massey scheme, a framework which combines both Substitution Permutation Network and Feistel Network features. Its resistance to the imprimitivity attack is obtained as a consequence of a more general result in which the problem of proving the primitivity of the Lai-Massey scheme is reduced to the simpler one of proving the primitivity of the group generated by the round functions of a strictly related Substitution Permutation Network.

cs.CR↗