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. https://ijgt.ui.ac.ir/article_30571.html Let \(G\) be a group, and let \(H \le K \le G\) be subgroups. We say that \(H\) is \emph{perfectly embedded} in \(K\) if \([H,K]=H\). In the special case \(K=G\), we simply say that \(H\) is a perfectly embedded subgroup of \(G\). The aim of this paper is to investigate several aspects of this notion. We first characterize the groups whose only perfectly embedded subgroup is the trivial subgroup, showing that they are exactly the hypocentral groups. At the other extreme, we study groups in which every normal subgroup is perfectly embedded, providing a complete classification of finite \(\mathcal T\)-groups with this feature. We also examine groups in which every subgroup is perfectly embedded in an appropriate subgroup normalizing it. Finally, attention is restricted to abelian normal subgroups, focusing once more on the two opposite situations. In particular, we show that every finite Frobenius group belongs to the class of groups in which every abelian normal subgroup is perfectly embedded. https://ijgt.ui.ac.ir/article_30572.html This paper investigates two problems concerning products of conjugacy classes and irreducible characters in finite groups. First, we provide a simplified proof of Dade and Yadav's theorem concerning groups in which the conjugacy class product $x^Gy^G=(xy)^G$ holds for all $x\in G, y\in G-(x^{-1})^G$. By using the classification of Camina groups due to Dark and Scoppola, we demonstrate that such groups are either abelian, non-abelian Camina $p$-groups, or specific Frobenius groups of the form $F^{+}\rtimes F^{\times}$ for some finite field $F$ with $|F|>2$ or $E_9\rtimes Q_8$. Second, we explore the dual problem for characters: groups in which the product of any two non-conjugate irreducible characters has exactly one irreducible constituent. We establish that such groups are precisely the Camina $p$-groups and the two families of Frobenius groups mentioned above. https://ijgt.ui.ac.ir/article_30674.html In the present paper, we investigate a general form of totally imprimitive finitary permutation groups. Totally imprimitive groups were originally defined by P. M. Neumann for subgroups of the finitary symmetric group acting on an infinite set. We aim to extend this definition to arbitrary subgroups of symmetric groups, which are not necessarily finitary. To this end, we introduce a property that we shall call the support-block property for subgroups of symmetric groups. This property allows us to generalize several results previously obtained for totally imprimitive finitary permutation groups to a broader class of permutation groups acting on infinite sets. We recover and extend various structural results obtained in earlier studies, with particular emphasis on transitive permutation groups. Moreover, we reconstruct an example arising in previous studies on finitary permutation groups and show that it satisfies the support-block property, illustrating the applicability of our approach. Several consequences and applications of this framework are also discussed. https://ijgt.ui.ac.ir/article_30681.html We prove a rigidity result for finite skew braces whose multiplicative group is the direct product of two finite non-abelian simple groups. More precisely, we show that if (B) is a finite skew brace with ((B,\cdot)\cong S\times T), where (S) and (T) are finite non-abelian simple groups, then the additive group ((B,+)) cannot be supersolvable. This result contributes to the study of the restrictions imposed by the multiplicative group on the additive group of a skew brace, a central question related to the Byott–Vendramin conjecture. The proof combines structural properties of supersolvable groups, the existence of characteristic Hall subgroups, and classification results on finite simple groups admitting subgroups of 2-power index. These tools reduce the possible multiplicative groups to groups of projective linear type and force the occurrence of a section isomorphic to (\operatorname{PSL}_2(7)). The remaining cases are excluded through an analysis of suitable Sylow subgroups and the induced lambda action.