University of IsfahanInternational Journal of Group Theory2251-765015320250708On the normalizer-solubilizer conjecture1071222963010.22108/ijgt.2025.143871.1938ENHamidMousaviDepartment of Mathematics,
University of Tabriz,
P.O.Box 51666-17766,
Tabriz, IranJournal Article20250101Let $G$ be a finite group and $x$ be an element of $G$. Define ${\rm Sol}_G(x)$ as the set of all $y \in G$ such that $\langle{x,y}\rangle$ is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely $|\mathcal{N}_G(\langle{x}\rangle)| \mid |{\rm Sol}_G(x)|$, where $\mathcal{N}_G(\langle{x}\rangle)$ is the normalizer of $\langle{x}\rangle$. Furthermore, we demonstrate that the conjecture holds in the special case where $\mathcal{N}_G(\langle{x}\rangle)$ is a Frobenius group with kernel $\mathcal{C}_G(x)$, the centralizer of $x$ and $|\mathcal{N}_G(\langle{x}\rangle): \mathcal{C}_G(x)|$ is of prime order. Finally, we will classify all finite simple groups $G$ that contain an element $x$ for which ${\rm Sol}_G(x)$ is a maximal subgroup of order $pq$, where $p$ and $q$ are prime numbers.https://ijgt.ui.ac.ir/article_29630_9bcf8863ef5426a3f803f018838306df.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765015320250930Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$1231342973310.22108/ijgt.2025.144154.1942ENGareth AneurinJonesSchool of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UKJournal Article20250201This work is motivated by results obtained and problems posed by Bianchi, Camina, Lewis, Pacifici and Sanus, counting conjugacy classes of non-self-normalising subgroups of finite groups. We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups $G={\rm PSL}_2(p)$, $p$ prime, and for the numbers of those conjugacy classes which do or do not consist of self-normalising subgroups. The formulae are used to prove lower bounds $17$, $18$, $6$ and $12$ respectively satisfied by these invariants for all $p>37$. A computer search carried out for a different but related problem shows that these bounds are attained for over a million primes $p$; we show that if the Bateman--Horn Conjecture is true, they are attained for infinitely many primes. Also, assuming no unproved conjectures, we use a result of Heath-Brown to obtain upper bounds for these invariants, valid for an infinite set of primes $p$.https://ijgt.ui.ac.ir/article_29733_7b7b72a5c9345aa24f7d8e0f154f880a.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765015320250918A new notion derived from the deep commuting graph1351442973510.22108/ijgt.2025.145983.1971ENBahramArvinDepartment of Mathematics, Faculty of Science, Razi University, Kermanshah, IranBehrouzEdalatzadehDepartment of Mathematics, Faculty of Science, Razi University, Kermanshah, IranAliRezaSalemkarFaculty of Mathematical Sciences, Shahid Beheshti
University, Tehran, IranJournal Article20250715The deep commuting graph of a given group $G$ is a graph whose vertex set is $G$, and two elements of $G$ are adjacent if their inverse images in every central extension of $G$ commute. In this paper, we introduce the concept of deep isoclinism for groups and tie this concept to the deep commuting graphs.https://ijgt.ui.ac.ir/article_29735_1e15ea8ff69e7ae8c991db56ffdbb569.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765015320250921Computational aspects of subindices and subfactors with characterization of finite index stable groups1451602978610.22108/ijgt.2025.142465.1919ENMohammad HadiHooshmandDepartment of Mathematics, Shi.C., Islamic Azad University, Shiraz, IranMoeinYousefian AraniDepartment of Mathematical Sciences, Sharif University of Technology, Azadi Avenue, Tehran, 11155-9415, Tehran, IranJournal Article20240814Recently, subindices and subfactors of groups with connections to number theory, additive combinatorics, and factorization of groups have been introduced and studied. Since all group subsets are considered in the theory and there are many basic open problems and questions, their computational aspects are of particular importance. In this paper, by introducing some computational methods and using theoretical approaches together, we not only solve several problems but also pave the way for studying the topic. As the most important result of the study, we completely characterize finite index stable groups.https://ijgt.ui.ac.ir/article_29786_047827727c732b96057b050b1ff4c9b2.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765015320251126A group with exactly one noncommutator1611673007810.22108/ijgt.2025.146052.1975ENOmar HatemSalemMathematics and Actuarial Science Department, The American University in Cairo, Cairo, EgyptDaoudSinioraMathematics and Actuarial Science Department, The American University in Cairo, Cairo, EgyptJournal Article20250730The 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. 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 Holt and Plesken’s list 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. We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this is the smallest order of such a group. 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.https://ijgt.ui.ac.ir/article_30078_67b173b978b85221b6ceb66220f43c68.pdf