arXiv subjects

M. Trombetti

Publications and source records attributed to M. Trombetti.

5 recordsLinked to original sources

arXiv2026

On the independence of permutation characters

Let $G$ be a finite group. For every subgroup $H\leq G$, let $\pi_H$ be the permutation character of the action of $G$ on the left cosets of $H$. We prove that the characters $\pi_H$, with $H$ running through representatives of the conjugacy classes of subgroups of $G$, are linearly independent if and only if $G$ is cyclic. In particular, no finite insoluble group has the property asked for in Kourovka Notebook Problem~11.9. The proof uses only the fixed-point formula for a coset action and an elementary triangular-matrix argument.

math.GR↗

arXiv2026

Hopficity of profinite completions of abelian groups

We determine exactly when the profinite completion of an arbitrary abelian group is topologically Hopfian. For an abelian group $A$, we prove that \[ \widehat A \text{ is topologically Hopfian} \quad\Longleftrightarrow\quad A/pA \text{ is finite for every prime }p. \] As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes $q_i$, the group $\bigoplus_{i\geq1}\Z[1/q_i]$ is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian.

math.GR↗

arXiv2026

The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces

In this short note we establish the Schur--Zassenhaus Theorem and Sylow's Third Theorem for finite skew braces. More precisely, we prove that every Hall ideal of a finite skew brace admits a sub-skew brace complement, and more generally that every left ideal whose order is coprime to that of the Hall ideal can be embedded in such a complement. Using similar ideas we show that every left ideal of prime-power order is contained in a Sylow sub-skew brace. Finally, we prove that the number of Sylow $p$-sub-skew braces is congruent to $1$ modulo $p$, and provide examples showing that the corresponding containment property fails for arbitrary sub-skew braces.

math.GR↗

arXiv2025

On Dedekind Skew Braces

Skew braces play a central role in the theory of set-theoretic non-degenerate solutions of the Yang--Baxter equation, since their algebraic properties significantly affect the behaviour of the corresponding solutions (see for example [Ballester-Bolinches et al., Adv. Math. 455 (2024), 109880]). Recently, the study of nilpotency-like conditions for the solutions of the Yang--Baxter equation has drawn attention to skew braces of abelian type in which every substructure is an ideal (so-called, Dedekind skew braces); see for example [Ballester-Bolinches et al., Result Math. 80 (2025), Article Number 21]. The aim of this paper is not only to show that the hypothesis the skew brace is of abelian type can be neglected in essentially all the known results in this context, but also to extend this theory to skew braces whose additive or multiplicative groups are locally cyclic (and more in general of finite rank). Our main results -- which are in fact much more general than stated here -- are as follows: (1) Every finite Dedekind skew brace is centrally nilpotent. (2) Every hypermultipermutational Dedekind skew brace with torsion-free additive group is trivial. (3) Characterization of a skew brace whose additive or multiplicative group is locally cyclic (4) If a set-theoretic non-degenerate solution of the Yang--Baxter equation has a Dedekind structure skew brace and fixes the diagonal elements, then such a solution must be the twist solution.

math.RA↗

arXiv2025

On the Sylow Theorem for Skew Braces

We discuss the (first) Sylow theorem for certain classes of finite skew braces, proving it to hold true when the skew brace is two-sided, bi-skew, right nilpotent, $\lambda$-homomorphic or supersoluble. We also show it to hold true for soluble skew braces that are left-nilpotent, and address a number of more specialized settings, proving general Hall-type theorems.

math.RA↗