https://ijgt.ui.ac.ir/ International Journal of Group Theory en daily 1 Tue, 01 Dec 2026 00:00:00 +0330 Tue, 01 Dec 2026 00:00:00 +0330 https://ijgt.ui.ac.ir/article_30123.html 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_30141.html 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_30136.html 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_30199.html 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_30198.html 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_30287.html In this paper, we study the structure of the unit group of the semisimple group algebra $F_q[SL(2,F_{7})]$, where $F_q$ is a finite field of characteristic $p \neq 2,3,7$, and $SL(2,F_{7})$ denotes the special linear group consisting of $2 \times 2$ matrices over the finite field $F_{7}$ with determinant 1. The group $SL(2,F_{7})$ is a finite non-abelian group of order 336, and its group algebra over a finite field provides an interesting case for algebraic exploration, especially in the semisimple case, which occurs when the characteristic of the field does not divide the order of the group. We provide a detailed characterization of the unit group $U(F_q[SL(2,F_{7})])$ and examine whether $SL(2,F_{7})$ has a normal complement within this unit group. Our findings show that for all primes $p \neq 2,3,7 $, the group $SL(2,F_{7})$ does not admit a normal complement in $U(F_q[SL(2,F_{7})])$. This contributes to the broader understanding of the subgroup structure and normality conditions in unit groups of semisimple group algebras involving non-abelian quasi-simple groups. https://ijgt.ui.ac.ir/article_30303.html The purpose of this paper is to explore possible definitions of $n$-isoclinism for skew braces. We also introduce the notions of verbal sub-skew braces and marginal left ideals.