arXiv subjects

Mario Galici

Publications and source records attributed to Mario Galici.

6 recordsLinked to original sources

arXiv2026

$2$-designs admitting a flag-transitive automorphism group with socle $PSL(2,q)$

$2$-designs admitting a flag-transitive automorphism group $G$ with socle $PSL(2,q)$, where $q=p^{f}\geq 4$, are investigated in both the point-primitive and point-imprimitive cases. In the latter case, a complete classification is achieved, and three known examples occur, namely: the complementary designs of $PG(3,2)$ and $PG(3,4)$, and the $2$-$(36,8,4)$ design constructed by Devillers and Praeger in [14]. In the point-primitive case, apart from the Witt-Bose-Shrikhande linear spaces of even order $q$, $48$ sporadic examples are classified. Surprisingly, one of these numerical examples is the linear space with $v=496$ and $k=4$ admitting $P\Gamma L(2,2^{5})$ as a flag-transitive automorphism group, which was missing in the 1990 classification by Buekenhout et al. [7,36,12].

math.CO↗

arXiv2025

$\mathrm{ EA}(q)$-additive Steiner 2-designs

A design is $G$-additive with $G$ an abelian group, if its points are in $G$ and each block is zero-sum in $G$. All the few known ``manageable" additive Steiner 2-designs are $\mathrm{EA}(q)$-additive for a suitable $q$, where $\mathrm{EA}(q)$ is the elementary abelian group of order $q$. We present some general constructions for $\mathrm{EA}(q)$-additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an additive $\mathrm{EA}(2^8)$-additive 2-$(52,4,1)$ design which is also resolvable, and three pairwise non-isomorphic $\mathrm{EA}(3^5)$-additive 2-$(121,4,1)$ designs, none of which is the point-line design of $\mathrm{PG}(4,3)$. In the attempt to find also an $\mathrm{EA}(2^9)$-additive 2-$(511,7,1)$ design, we prove that a putative 2-analog of a 2-$(9,3,1)$ design cannot be cyclic.

math.CO↗

arXiv2025

The Flag-Transitive and Point-Imprimitive Symmetric $(v,k,\lambda)$ Designs with $v<100$

A complete classification of the flag-transitive point-imprimitive symmetric $2$-$(v,k,\lambda )$ designs with $v<100$ is provided. Apart from the known examples with $\lambda \leq 10$, the complementary design of $PG_{5}(2)$, and the $2$-design $\mathcal{S}^{-}(3)$ constructed by Kantor in \cite{Ka75}, we found two non isomorphic $2$-$(64,28,12)$ designs. They were constructed via computer as developments of $(64,28,12)$-difference sets by AbuGhneim in \cite{OAG}. In the present paper, independently from \cite{OAG}, we construct the aforementioned two $2$-designs and we prove that their full automorhpism group is flag-transitive and point-imprimitive. The construction is theoretical and relies on the the absolutely irreducible $8$-dimensional $\mathbb{F}_{2}$-representation of $PSL_{2}(7)$. Our result, together with that about the flag-transitive point-primitive symmetric $2$-designs with $v<2500$ by Brai\'{c}-Golemac-Mandi\'{c}-Vu\v{c}i\v{c}i\'{c} \cite{BGMV}, provides a complete classification of the flag-transitive $2$-designs with $v<100$.

math.GR↗

arXiv2025

Extensions of Steiner Triple Systems

In this article we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.

math.CO↗

arXiv2025

On the number of small Steiner triple systems with Veblen points

The concept of Schreier extensions of loops was introduced in the general case in [11] and, more recently, it has been explored in the context of Steiner loops in [6]. In the latter case, it gives a powerful method for constructing Steiner triple systems containing Veblen points. Counting all Steiner triple systems of order v is an open problem for v>21. In this paper, we investigate the number of Steiner triple systems of order 19, 27 and 31 containing Veblen points and we present some examples.

math.CO↗

arXiv2024

An extension formula for right Bol loops arising from Bol reflections

We study a new extension formula for right Bol loops. We prove the necessary or sufficient conditions for the extension to be right Bol. We describe the most important invariants: right multiplication group, nuclei, and center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. We also derive further results on the structure group of the core of the extension.

math.GR↗