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Daoud Siniora

Publications and source records attributed to Daoud Siniora.

8 recordsLinked to original sources
arXiv2026

Shirshov's amalgamated free product and generic nilpotent groups

In earlier work, the authors gave a construction and description of an amalgamated free product of filtered Lie algebras within a fixed nilpotency class, based on an intricate induction rooted in the work of Maier and of Higman. In this paper, the authors give a new description of this amalgam, adopting the viewpoint and methods from the work of A. I. Shirshov on amalgamation of Lie algebras, substantially simplifying their previous approach. This new description yields both group-theoretic and descriptive set-theoretic applications. A $c$-nilpotent group is called UL-equivalent if its lower and upper central series coincide. We prove that every finite $c$-nilpotent group of prime exponent $p$ with $p>c$ embeds in a finite UL-equivalent $c$-nilpotent group of exponent $p$. This recovers a result due to Ivanov and Majcher that the Polish space of enumerated $c$-nilpotent groups of exponent $p>c$ has a comeager orbit. Our result also has the following consequences. First, the class of finite $c$-nilpotent groups of exponent $p>c$ has the \textit{cofinal} amalgamation property (answering a question of Ivanov and Majcher, who showed that it has the \textit{weak} amalgamation property). Second, the reduct of the Fra\"iss\'e limit of $c$-Lazard groups of exponent $p>c$ to the group language is generic in the space of enumerated groups. Finally, we prove analogues of these results also for torsion-free $c$-nilpotent groups.

math.GR↗
arXiv2026

Cofinality of Regular Tournaments

We show that the class of all finite regular tournaments is cofinal in the class of finite tournaments. In addition, we establish cofinality results for certain special subclasses of regular tournaments. We also provide an algorithm for constructing these regular tournaments.

math.CO↗
arXiv2025

A Group with Exactly One Noncommutator

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.

math.GR↗
arXiv2024

A two-sorted theory of nilpotent Lie algebras

We prove the existence of a model companion of the two-sorted theory of $c$-nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field sort. Using a new criterion which does not rely on a stationary independence relation, we prove that if the field is NSOP$_1$, then the model companion is NSOP$_4$. We also prove that if the field is algebraically closed, then the model companion is $c$-NIP.

math.LO↗
arXiv2023

Model-theoretic properties of nilpotent groups and Lie algebras

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fra\"iss\'e limit of 2-nilpotent groups of exponent $p$ studied by Baudisch is 2-dependent and NSOP$_{1}$. We prove that the class of $c$-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for $2 < c$, the generic $c$-nilpotent Lie algebra over $\mathbb{F}_{p}$ is strictly NSOP$_{4}$ and $c$-dependent. Via the Lazard correspondence, we obtain the same result for $c$-nilpotent groups of exponent $p$, for an odd prime $p > c$.

math.LO↗
arXiv2019

Coherent extension of partial automorphisms, free amalgamation, and automorphism groups

We give strengthened versions of the Herwig-Lascar and Hodkinson-Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraisse classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraisse limit of any Fraisse class that is the class of all $\mathcal{F}$-free structures (in the Herwig--Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language, all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

math.LO↗