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Marco Trombetti

Publications and source records attributed to Marco Trombetti.

18 recordsLinked to original sources
arXiv2026

A probabilistic approach to the Yang--Baxter equation and skew braces

We investigate finite non-degenerate set-theoretic solutions to the Yang--Baxter equation and skew braces using a probabilistic approach. We introduce four probabilities that measure how far a solution is from being a flip, but in different ways. Our main results state that for solutions arising from skew braces, these probabilities exhibit a rigid behaviour --- apart from a finite list of exceptional values (which we show to occur by means of explicit examples), they admit an upper bound that is slightly above $\frac{1}{2}$. In the skew brace setting, these probabilities measure how far the underlying skew brace is from being a trivial brace (in two different ways: one via the annihilator and the other via the socle), a trivial skew brace, and an almost trivial skew brace. We also introduce a probability that is related to the indecomposable components of a solution, and a probability that measures how close an arbitrary bijective non-degenerate map is to being a solution. In contrast to our main results, these two probabilities do not exhibit a discrete behaviour near $1$ --- they can be made arbitrarily close to $1$.

math.GR↗
arXiv2026

A torsion-free group of nilpotency class six with all subgroups subnormal of defect at most $5$

Casolo asked whether a torsion-free group in which every subgroup is subnormal of defect at most $n$ must be nilpotent of class at most $n$. The answer is known to be positive for $n\leq 4$. We construct a $2$-generated torsion-free nilpotent group $G$ such that $$ \cl(G)=6 \qquad\text{and}\qquad [G,{}_5H]\leq H \quad\text{for every }H\leq G. $$ Thus Casolo's question has a negative answer for $n=5$.

math.GR↗
arXiv2026

A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group

We give a negative solution to Problem~13.23 of the Kourovka Notebook. We construct a torsion-free group $G$ of Hirsch length $14$ admitting a finite series \[ 1=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_{14}=G \] in which every $G_i$ is normal in $G$ and every factor is infinite cyclic, but such that $\Out(G)=1$.

math.GR↗
arXiv2026

Free Skew Braces and Free Solutions of the Yang--Baxter Equation

We offer a workable construction of the free right nilpotent skew braces of arbitrary class which allows us to prove (among many other things) that this free object has free additive/multiplicative groups, and that it must also be residually finite and Hopfian. We introduce the class of right nilpotent solutions, which correspond to right nilpotent skew braces. As a consequence of our construction, the free solutions in this class have a solvable Word Problem, and every law holding for finite solutions of the previous type also holds for every solution of the same type. In the remainder of the paper, we present further explicit realizations of free objects and explore their consequences. Among these are free two-sided skew braces of abelian type (with an abelian multiplicative group) and free centrally nilpotent skew braces of class 2.

math.GR↗
arXiv2026

Finite groups with a large normalized sum of element orders

For a finite group $G$, let $\psi(G)$ be the sum of the orders of its elements, and define the corresponding normalized sum as $\psi'(G) := \psi(G)/\psi(\mathcal{C}_{|G|})$, where $\mathcal{C}_{|G|}$ is the cyclic group of the same order as $G$. Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if $\psi'(G)>\psi'(D_8) = \frac{19}{43}$, then $G$ belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. Moreover, the equality case is fully settled. Finally, our arguments lead to a complete description of all groups satisfying $\psi'(G)> \psi'(A_4) = \frac{31}{77}$, thereby fully determining the groups covered by the supersolubility criterion of Baniasad Azad and Khosravi [Canad. Math. Bull. 65 (2022), 30--38], and thus providing a more complete answer to a corresponding conjecture of T\v{a}rn\v{a}uceanu.

math.GR↗
arXiv2025

The Pseudocentre of a Group (with an appendix by Anthony Genevois)

In 1973, Jim Wiegold introduced the concept of pseudocentre P(G) of a group G as the intersection of the normal closures of the centralizers of its elements. He proved that the pseudocentre of a non-trivial finite group is always non-trivial, giving a new variable on which one can use induction in finite group theory. In the same paper, Wiegold states that no obvious relations seem to hold between the pseudocentre and the canonical characteristic subgroups of a group. The aim of this work is to show that the pseudocentre is indeed much more involved in the structure of an arbitrary group then anyone could have expected. For example, we prove that a soluble group coincides with its pseudocentre if and only if it is abelian, and that the structure of the commutator subgroup strongly influences the structure of the pseudocentre. And this is not the end of the story. In fact, the behaviour of the pseudocentre in arbitrary (possibly infinite) groups can be extremely wild: sometimes it is very difficult even to understand whether the pseudocentre is trivial or not. This wilderness is exampled by some of our main results (see the introduction for a complete list): 1) There exists a polycyclic group of Hirsch length 3 in which the pseudocentre is trivial. 2) The pseudocentre of the group of unitriangular matrices over any field is the largest term of the upper central series that is abelian. 3) Free products have a trivial pseudocentre, but there exist amalgamated free products of non-trivial groups coinciding with their pseudocentre. 4) Weakly regular branch groups have a trivial pseudocentre. 5) The pseudocentre of the Thompson group is the derived subgroup. 6) Wreath products can have a totally arbitrary pseudocentre.

math.GR↗
arXiv2025

On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation

The aim of this paper is to provide purely arithmetical characterisations of those natural numbers $n$ for which every non-degenerate set-theoretic solution of cardinality $n$ of the Yang--Baxter equation arising from a skew brace (sb-solution for short) satisfies some relevant properties, such as being a flip or being involutive. For example, it turns out that every sb-solution of cardinality $n$ has finite multipermutation level if and only if its prime factorisation $n= p_1^{\alpha_1} \ldots p_t^{\alpha_t}$ is cube-free, namely $\alpha_i\leq 2$ for every $i$, and $p_i$ does not divide $p_j^{\alpha_j}-1$ for $i\neq j$. Two novel constructions of skew braces will play a central role in our proofs. We shall also introduce the notion of supersoluble solution and show how this concept is related to that of supersoluble skew brace. In doing so, we have spotted an irreparable mistake in the proof of Theorem C [Ballester-Bolinches et al., Adv. Math. 455 (2024)], which characterizes soluble solutions in terms of soluble skew braces.

math.GR↗
arXiv2025

Skew Braces from a model-theoretic point of view 1

Skew braces are one of the main algebraic tools controlling the structure of a non-degenerate bijective set-theoretic solution of the Yang-Baxter equation. The aim of this paper is to study model-theoretically tame skew braces, with particular attention to the notions of solubility and nilpotency.

math.GR↗
arXiv2024

Follow the money: a startup-based measure of AI exposure across occupations, industries and regions

The integration of artificial intelligence (AI) into the workplace is advancing rapidly, necessitating robust metrics to evaluate its tangible impact on the labour market. Existing measures of AI occupational exposure largely focus on AI's theoretical potential to substitute or complement human labour on the basis of technical feasibility, providing limited insight into actual adoption and offering inadequate guidance for policymakers. To address this gap, we introduce the AI Startup Exposure (AISE) index-a novel metric based on occupational descriptions from O*NET and AI applications developed by startups funded by the Y Combinator accelerator. Our findings indicate that while high-skilled professions are theoretically highly exposed according to conventional metrics, they are heterogeneously targeted by startups. Roles involving routine organizational tasks-such as data analysis and office management-display significant exposure, while occupations involving tasks that are less amenable to AI automation due to ethical or high-stakes, more than feasibility, considerations -- such as judges or surgeons -- present lower AISE scores. By focusing on venture-backed AI applications, our approach offers a nuanced perspective on how AI is reshaping the labour market. It challenges the conventional assumption that high-skilled jobs uniformly face high AI risks, highlighting instead the role of today's AI players' societal desirability-driven and market-oriented choices as critical determinants of AI exposure. Contrary to fears of widespread job displacement, our findings suggest that AI adoption will be gradual and shaped by social factors as much as by the technical feasibility of AI applications. This framework provides a dynamic, forward-looking tool for policymakers and stakeholders to monitor AI's evolving impact and navigate the changing labour landscape.

econ.GN↗
arXiv2024

Finite skew braces of square-free order and supersolubility

The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers, and that in an arbitrary supersoluble skew brace $B$ many relevant skew brace-theoretical properties are easier to identify: for example, a centrally nilpotent ideal of $B$ is $B$-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, $B$ has finite multipermutational level if and only if $(B,+)$ is nilpotent. Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite non-degenerate solution of the Yang--Baxter Equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

math.GR↗
arXiv2024

On the lattice of closed subgroups of a profinite group

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behaviour of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.

math.GR↗
arXiv2023

On derived-indecomposable solutions of the Yang--Baxter equation

If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.

math.GR↗
arXiv2023

A note on right-nil and strong-nil skew braces

The aim of this short note is to completely answer Questions 2.34 and 2.35 of arXiv:1806.01127. In particular, we show that a finite strong-nil skew brace $B$ of abelian type need not be right-nilpotent, but that this is the case if~$B$ is of nilpotent type and $b\ast b=0$ for all $b\in B$ (our examples show that this is the best possible result).

math.GR↗
arXiv2023

Central nilpotency of left skew braces and solutions of the Yang-Baxter equation

Nipotency of skew braces is related to certain types of solutions of the Yang-Baxter equation. This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory (Section 4.1) and we introduce an "index" for subbraces (Section 4.2), but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent (Example B), a rather peculiar fact. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a {\it good} Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient in our work is the characterisation of the commutator of ideals in terms of absorbing polynomials (Section 3); this solves Problem 3.4 of arXiv:2109.04389. Moreover, we provide an example (Example A) showing that the idealiser of a subbrace (as defined in arXiv:2205.01572v2) does not exist in general.

math.GR↗
arXiv2023

Joins of $σ$-subnormal subgroups

Let $σ=\{σ_j\,:\, j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup $X$ of a finite group $G$ is~\textit{$σ$-subnormal} in $G$ if there exists a chain of subgroups $$X=X_0\leq X_1\leq\ldots\leq X_n=G$$ such that, for each $1\leq i\leq n-1$, $X_{i-1}\trianglelefteq X_i$ or $X_i/(X_{i-1})_{X_i}$ is a $σ_{j_i}$-group for some $j_i\in J$. Skiba~[12] studied the main properties of $σ$-subnormal subgroups in finite groups and showed that the set of all $σ$-subnormal subgroups plays a relevant role in the structure of a finite soluble group. In [5], we laid the foundation of a general theory of $σ$-subnormal subgroups (and $σ$-series) in locally finite groups. It turns out that the main difference between the finite and the locally finite case concerns the behaviour of the join of $σ$-subnormal subgroups: in finite groups, $σ$-subnormal subgroups form a sublattice of the lattice of all subgroups [3], but this is no longer true for arbitrary locally finite groups. This is similar to what happens with subnormal subgroups, so it makes sense to study the class $\mathfrak{S}_σ^\infty$ (resp. $\mathfrak{S}_σ$) of locally finite groups in which the join of (resp. of finitely many) $σ$-subnormal subgroups is $σ$-subnormal. Our aim is to study how much one can extend a group in one of these classes before going outside the same class (see for example Theorems~3.6, 3.8, 5.5 and 5.7). Also, $σ$-subnormality criteria for the join of $σ$-subnormal subgroups are obtained: similarly to a celebrated theorem of Williams (see [15]), we give a necessary and sufficient conditions for a join of two $σ$-subnormal subgroups to always be $σ$-subnormal; consequently, we show that the join of two orthogonal $σ$-subnormal subgroups is $σ$-subnormal (extending a result of Roseblade [11]).

math.GR↗
arXiv2022

A note on the series' of ordinal numbers

The aim of this short note is to provide a proof to a statement of Sierpiński concerning the number of possible sums of a series (of type $λ<\aleph_1$) of arbitrary ordinal numbers.

math.LO↗