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Marco Damele

Publications and source records attributed to Marco Damele.

13 recordsLinked to original sources
arXiv2026

Ideals and Solvability in Skew Braces

We investigate how nilpotency assumptions on the multiplicative group of a finite skew brace constrain its ideal structure and solvability. Our first main result shows that, if \(B=(B,+,\cdot)\) is finite and \((B,\cdot)\) is nilpotent, then the additive Fitting subgroup \(F(B,+)\) is a non-zero ideal of \(B\). As consequences, every finite simple skew brace with nilpotent multiplicative group is isomorphic to \(\Triv(C_p)\) for some prime \(p\), and every such skew brace admits an ideal of prime index. In particular, \[ B*B\neq B \qquad\text{and}\qquad \partial(B) \neq B. \] We also show that nilpotency of the multiplicative group does not, in general, imply either left nilpotency or solvability. Motivated by this obstruction, we then study the solvability of two-sided skew braces, both in the finite and in the general setting. We prove that, whenever \(I\) is an ideal of a two-sided skew brace \(B\), the internal commutator ideal \([I,I]_I\) is again an ideal of \(B\). This yields an extension theorem for solvability and implies that, for finite two-sided skew braces, solvability of the skew brace is equivalent to solvability of either the additive or the multiplicative group. In particular, every finite skew brace with abelian multiplicative group is solvable. Finally, we show that, although this equivalence fails in general for infinite two-sided skew braces, a residual form of it still survives: if the additive group is solvable, then every finite homomorphic image of the multiplicative group is solvable.

math.GR↗
arXiv2026

Simple Skew Braces with Cyclic Sylow Subgroups

We study simplicity and splitting phenomena in finite skew braces under cyclicity assumptions on Sylow subgroups. We first classify finite simple skew braces whose multiplicative group is a \(Z\)-group. If the additive group is soluble, then the skew brace is either trivial of prime order or isomorphic to one of the two simple skew braces of order \(12\) with additive group \(A_4\) and multiplicative group \(C_3\rtimes C_4\). If the additive group is insoluble, then it is necessarily isomorphic to \(\operatorname{PSL}_2(p)\) for some prime \(p\geq5\). This conclusion is sharp, since such examples exist for every prime \(p\geq5\). We then consider the more general situation in which only a Sylow subgroup corresponding to the smallest prime divisor \(p\) of the order is assumed to be cyclic. Under suitable hypotheses on the additive or multiplicative Sylow \(p\)-subgroup, we prove that the skew brace contains a Hall \(p'\)-ideal and splits as a semidirect product of this ideal with a Sylow \(p\)-subbrace. As a consequence, every finite simple skew brace satisfying one of these hypotheses is trivial of prime order. Moreover, as a consequence of our splitting theorem, we verify Byott's solvability conjecture for finite skew braces whose additive group has a cyclic Sylow $2$-subgroup.

math.GR↗
arXiv2026

A Schur--Zassenhaus Theorem for Finite Skew Braces

We prove a Schur--Zassenhaus theorem for finite skew braces. More precisely, if \(B\) is a finite skew brace and \(I\) is an ideal of \(B\) such that \(|I|\) and \(|B/I|\) are coprime, then \(I\) admits a complement in \(B\).

math.GR↗
arXiv2026

On Simply Connected Simple Lie Skew Braces with Nilpotent Multiplicative Group

We prove that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if $(G,\cdot,\circ)$ is a simply connected Lie skew brace with nilpotent multiplicative Lie group and $\dim G>1$, then $(G,\cdot,\circ)$ is not simple. Thus, in the simply connected Lie setting, nilpotency of the multiplicative group is incompatible with simplicity in every dimension greater than one. The proof is carried out at the post-Lie algebra level. First, if the additive Lie algebra is solvable, then its nilradical is automatically an ideal of the associated post-Lie algebra. Second, when both Lie algebras underlying an integrable post-Lie structure are nilpotent, one always obtains a proper post-Lie ideal with trivial quotient. To pass from infinitesimal ideals to global ideals of the Lie skew brace, we show that trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals.

math.GR↗
arXiv2026

On finite perfect two-sided skew braces

We prove a structure theorem for finite perfect two-sided skew braces. The main tool is a central product theory for skew braces, developed here in both external and internal form; we show that these two constructions are equivalent. Our main result states that every finite perfect two-sided skew brace \(B\) admits the canonical decomposition $B=B^2\circ B^{2,\operatorname{op}},$ where \(B^2\) is almost trivial with perfect additive group, while \(B^{2,\operatorname{op}}\) is trivial with perfect additive group. Thus finite perfect two-sided skew braces are classified, up to central amalgamation, by trivial and almost trivial skew braces arising from perfect groups. This decomposition has strong consequences for the underlying groups: for finite two-sided skew braces, perfectness of the skew brace is equivalent to perfectness of either the additive or the multiplicative group. In the trivial-center case the central product becomes a direct product, recovering Trappeniers' classification of finite simple two-sided skew braces. We also show that quasi-simple two-sided skew braces are necessarily either trivial or almost trivial. Finally, we prove that this rigidity is genuinely two-sided by constructing a quasi-simple skew brace which is not two-sided and is neither trivial nor almost trivial.

math.GR↗
arXiv2026

Solvability and Rigidity for Topological Skew Braces

We study compact and locally compact topological analogues of the Byott--Vendramin solvability problem for finite skew braces, asking whether solvability of the additive group forces solvability of the multiplicative group. Our main theorem proves an affirmative result in the connected locally compact Hausdorff setting: if \(B=(B,\cdot,\circ)\) is a connected locally compact Hausdorff topological skew brace and the additive group \((B,\cdot)\) is solvable, then the multiplicative group \((B,\circ)\) is solvable. The proof proceeds by reducing the additive group to a solvable Lie quotient and then applying an affine-action theorem: a connected Lie group acting transitively and affinely on a connected solvable Lie group, with solvable stabilizer identity component, is itself solvable. We further show that the Hausdorff, local compactness, and connectedness hypotheses are essential by constructing counterexamples when each is omitted. In the compact connected Hausdorff case with abelian additive group, we obtain a stronger rigidity phenomenon: the two group laws coincide.

math.GR↗
arXiv2026

On simple compact Lie skew braces

We study simplicity of Lie skew braces from both global and infinitesimal perspectives. After reviewing the correspondence between connected Lie skew braces, simply transitive affine actions, and post-Lie algebras, we investigate ideals and rigidity phenomena. Our main result concerns compact connected Lie skew braces. We prove that any compact connected simple Lie skew brace is either the trivial Lie skew brace on \(S^1\), or both of its underlying Lie groups are simple and the brace is trivial or almost trivial. Consequently, apart from the exceptional \(S^1\) case, simplicity of a compact connected Lie skew brace is equivalent to simplicity of either underlying Lie group. We also show that every connected compact solvable Lie skew brace is trivial. Finally, we construct a noncompact example demonstrating that this rigidity phenomenon does not hold in general: there exists a connected simply connected simple Lie skew brace whose additive and multiplicative Lie groups are both solvable.

math.GR↗
arXiv2026

On the multiplicative group of a two-sided skew brace of solvable type

We prove that if $(B,+,\cdot)$ is a two-sided skew brace whose additive group is solvable, then every finite quotient of the multiplicative group $(B,\cdot)$ is solvable. In particular, our result recovers Nasybullov's theorem in the finite case ~\cite[Theorem~4.3(1)]{Nas} and extends it to arbitrary two-sided skew braces of solvable type.

math.GR↗
arXiv2026

Finite skew braces whose additive group is a Z-group

Rump proved in \cite[Theorem~1]{Rump2018ClassificationOC} that if a finite skew brace has cyclic additive group, then its multiplicative group is solvable and almost Sylow cyclic. In this paper we show that this rigidity persists when the additive group is a \(Z\)-group. More precisely, we prove that if \(B\) is a finite skew brace whose additive group is a \(Z\)-group, then \((B,\cdot)\) is solvable and almost Sylow cyclic. In addition, we show that every such skew brace is supersolvable; in particular, \((B,\cdot)\) is \(2\)-nilpotent. This extends \cite[Theorem~3.8]{ballesterbolinches2024finiteskewbracessquarefree} and recovers, in this broader setting, another result of Rump \cite[Proposition 13]{Rump2018ClassificationOC}. Finally, we prove that for skew braces of odd order the additive group is a \(Z\)-group if and only if the multiplicative group is a \(Z\)-group.

math.GR↗
arXiv2026

On a Cauchy theorem for finite skew braces

One of the major problems in the structural theory of skew braces consists in the classification of skew braces of finite order up to isomorphism. In this light, the open question of the existence of a Cauchy theorem for finite skew braces is of great interest. We prove a positive answer for the classes of finite two-sided skew braces and bi-skew braces. Consequences and related structural results are also outlined.

math.GR↗
arXiv2025

Finite groups in which every proper characteristic subgroup is cyclic

Let $G$ be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of $G$ are cyclic. We call such a group $\mathrm{CCS}$ group, short for \emph{Characteristic Cyclic}. In this paper, we provide a complete classification of these groups. As an application of our main result, we also make some progress toward the classification of minimal non-cyclic skew braces.

math.GR↗
arXiv2025

Structural and rigidity properties of Lie skew braces

We investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same identity element. LSBs extend skew left braces, which are central to the study of non-involutive set-theoretic solutions of the Yang--Baxter equation, to the smooth category. Our first main result shows that, for every connected LSB $(G,\cdot,\circ)$, linearity (in the simply-connected case) and solvability carry over from $(G,\cdot)$ to $(G,\circ)$, whereas the converse direction is rigid: if $(G,\circ)$ is nilpotent (respectively, semisimple) then $(G,\cdot)$ is forced to be solvable (respectively, isomorphic to $(G,\circ)$). Our second results provides two ``flexibility'' statements: every non-linear simply connected Lie group \((G, \cdot )\) admits an LSB \((G,\cdot,\circ)\) such that \((G,\circ)\) is linear, and every simply connected solvable Lie group \((G, \circ )\) supports a LSB \((G,\cdot,\circ)\) such that \((G,\cdot)\) is nilpotent. A third result provides a complete existence table for non-trivial LSBs across the six standard Lie-group classes, abelian, nilpotent (non-abelian), solvable (non-nilpotent), simple, semisimple (non-simple) and mixed type, identifying precisely when a LSB can be built and when only the trivial or no structure occurs. Both the explicit constructions and the properties established in our theorems rely on a factorisation technique for Lie groups, on the correspondence between LSBs and regular subgroups of the affine group $\operatorname{Aff}(G,\cdot)$, which renders LSB theory equivalent to simply transitive affine actions, and on the theory of post-Lie algebras together with their integrability properties.

math.GR↗