University of Isfahan International Journal of Group Theory 2251-7650 15 4 2026 12 01 Finite subgroups of automorphisms of free products 169 178 30123 10.22108/ijgt.2025.143507.1934 EN Ioannis Papavasileiou Independent Researcher, Athens, Greece Dionysios Syrigos Independent Researcher, Basingstoke, UK Journal Article 2024 11 26 We study finite subgroups of outer automorphisms of free products. We give upper bounds for the orders of these finite subgroups as well as bounds for the orders of individual torsion outer automorphisms under some (necessary) conditions for the free factors.

https://ijgt.ui.ac.ir/article_30123_1eaa7ef72954bb817776875438c29054.pdf

University of Isfahan International Journal of Group Theory 2251-7650 15 4 2026 12 17 Problems on Brauer characters 179 192 30141 10.22108/ijgt.2025.147244.1995 EN Yanjun Liu School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, China Wolfgang Willems Fakultat fur Mathematik, Otto-von-Guericke Universitat, Magdeburg, Germany and Departamento de Matematicas Universidad del Norte, Barranquilla, Colombia Jiping Zhang School of Mathematical Sciences, Peking University, Beijing, China Journal Article 2025 10 28 In this paper we focus on problems of irreducible $p$-Brauer characters and state conjectures based on many examples which we computed. Many of the asked questions hold true for $p$-solvable groups, but their answers in general seem to require a much deeper understanding than we have at the moment. The questions are dealing with degrees, Hilbert divisors, divisibility, height-zero irreducible Brauer characters, number of irreducible Brauer characters in a $p$-block, and Cartan invariants. For instance, one of the main conjectures states that an irreducible Brauer character has Hilbert divisor 1 if and only if the character lies in a $p$-block of defect zero. This is true for $p$-solvable groups since in this case there is a relation between Hilbert divisors and vertices. However, even for a $p$-block of a non-$p$-solvable group containing only two irreducible Brauer characters we do not have an idea how to attack the problem. We hope that the conjectures and questions which we state in this paper will inspire further research.

https://ijgt.ui.ac.ir/article_30141_b4f71d4959352b7f4a42f1421635ff44.pdf

University of Isfahan International Journal of Group Theory 2251-7650 15 4 2025 12 27 Hamiltonian degree of finite groups 193 206 30136 10.22108/ijgt.2025.144951.1956 EN Ahmad Erfanian Department of Pure Mathematics and Center of Exellence in Analysis on Algebraic Structures, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran Madeleine Al Tahan Department of Mathematics and Statistics, Abu Dhabi University, Abu Dhabi, United Arab Emirates Saba Al-Kaseasbeh Department of Mathematics, Tafila Technical University, Tafila, Jordan Journal Article 2025 04 16 This paper examines the probability of finite groups being Hamiltonian,a property defined by all subgroups being normal, and its implications for group structure analysis. To this end, we introduce the Hamiltonian degree, a novel extension of commutativity degrees in finite groups, and propose a comprehensive framework for its evaluation. Explicit formulas are derived for the Hamiltonian degree of dihedral groups, alongside general bounds applicable to broader group classes. Additionally, we explore its relationship to conjugacy class subgroups, shedding light on new structural connections within group theory.

https://ijgt.ui.ac.ir/article_30136_af8367e77faf9ebfdc60548f647561cc.pdf

University of Isfahan International Journal of Group Theory 2251-7650 15 4 2026 01 27 On characterization of simple $K_3$-groups by the number of elements of prime order 207 214 30199 10.22108/ijgt.2026.144924.1955 EN Lin Liu School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, P. R. China Yanheng Chen School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, P. R. China Songfang Jia School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, P. R. China Journal Article 2025 04 16 Since the classification theorem of finite simple groups was declared proven in the early 1980s, many group theorists have been attempting to delve deeper into the structure of simple groups from the perspective of group invariants, resulting in a series of research topics on the quantitative characterization of simple groups, such as spectral characterization, two-order characterization, and OD-characterization. In 2018, Moret\'{o} proposed a new conjecture for the characterization of finite simple groups by the group order and the number of elements of the largest prime order. A specific group whose order is divisible by exactly three distinct prime numbers is called a simple $K_3$-group. These groups form a simple class of finite non-abelian simple groups. This paper establishes a characterization of $L_2(8)$ and $L_3(3)$ by combining the group order with the number of elements of the largest prime order, which shows that the conjecture holds for all simple $K_3$-groups except $L_2(7)$, $U_3(3)$ and $U_4(2)$. In addition, we also characterize $L_2(7)$, $U_3(3)$ and $U_4(2)$ under additional condition of non-solvability. Furthermore, we prove that a conjecture of Li and Shi holds for the alternating groups $A_8$, $A_{10}$, and $L_2(7)$. Thus the conjecture of Li and Shi is valid for sporadic simple groups, for alternating groups $A_n(n\geq5)$, and for all simple $K_3$-groups except $U_3(3)$ and $U_4(2)$.

https://ijgt.ui.ac.ir/article_30199_639f23bdf03eda2ee8a472ce1702f0e1.pdf

University of Isfahan International Journal of Group Theory 2251-7650 15 4 2026 02 01 On Rota--Baxter operators on finite simple groups of Lie type 215 225 30198 10.22108/ijgt.2026.147698.2003 EN Alexey Galt Novosibirsk State University, Pirogova str. 1, P.O.Box 630090, Novosibirsk, Russia Vsevolod Gubarev Novosibirsk State University, Pirogova str. 1, P.O.Box 630090, Novosibirsk, Russia Journal Article 2025 12 05 Rota--Baxter operators on groups were introduced by L. Guo, H.~Lang, Yu.~Sheng in 2020. In 2023, V. Bardakov and the second author showed that all Rota--Baxter operators on simple sporadic groups are splitting, i.\,e. they correspond to exact factorizations of groups. In 2024, the authors of the current paper described all non-splitting Rota--Baxter operators on alternating groups. Now we describe Rota--Baxter operators on finite simple exceptional groups of Lie type and projective special linear groups of degree two.

https://ijgt.ui.ac.ir/article_30198_f2ea30188dc62982b146b62fd948eac5.pdf