University of IsfahanInternational Journal of Group Theory2251-765012120230301Proceedings of the Ischia Group Theory (2020/2021)27067ENJournal Article20221107https://ijgt.ui.ac.ir/article_27067_72fd2f3d9c5d9be26a1a00ee3353bcf1.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765012120230301On the automorphism groups of some Leibniz algebras1202605010.22108/ijgt.2021.130057.1735ENLeonid A.KurdachenkoDepartment of Geometry and Algebra, Oles Honchar Dnipro National University, 72 Gagarin Ave., Dnipro, UkraineAleksand A.PypkaDepartment of Geometry and Algebra, Oles Honchar Dnipro National University, 72 Gagarin Ave., Dnipro, UkraineIgor Y.SubbotinDepartment of Mathematics and Natural Sciences, College of Letters and Sciences,
National University, USAJournal Article20210815We study the automorphism groups of finite-dimensional cyclic Leibniz algebras. In this connection, we consider the relationships between groups, modules over associative rings and Leibniz algebras.https://ijgt.ui.ac.ir/article_26050_4e57796bac3dde4ac51ae723936e708d.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765012120230301On infinite anticommutative groups21262612110.22108/ijgt.2021.130377.1739ENCostantinoDeliziaDepartment of Mathematics, University of Salerno, ItalyChiaraNicoteraDepartment of Mathematics - University of Salerno - Via Giovanni Paolo II, 132 - 84084 - Fisciano (SA) ITALYJournal Article20210901We completely describe the structure of locally (soluble-by-finite) groups in which all abelian subgroups are locally cyclic. Moreover, we prove that Engel groups with the above property are locally nilpotent.https://ijgt.ui.ac.ir/article_26121_a8cf35139155ac9b06e596af2f15487a.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765012120230301Restrictions on sets of conjugacy class sizes in arithmetic progressions27342628810.22108/ijgt.2022.130654.1743ENAlan R.CaminaSchool of Mathematics, University of East Anglia Norwich, NR4 7TJ, UKRachel D.CaminaFitzwilliam College, Cambridge, CB3 0DG, UKJournal Article20210920We continue the investigation, that began in [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, <em>Ischia group theory 2010, World Sci. Publ.</em>, Hackensack, NJ (2012) 20--25.] and [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, <em>J. Group Theory</em>, <strong>23</strong> no. 6 (2020) 1039--1056.], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm cs}(G)$. Finite groups satisfying ${\rm cs}(G) = \{1, 2, 4, 6\}$ and $\{1, 2, 4, 6, 8\}$ are classified in [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, <em>J. Group Theory</em>, <strong>23</strong> no. 6 (2020) 1039--1056.] and [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, <em>Ischia group theory 2010, World Sci. Publ.</em>, Hackensack, NJ (2012) 20--25.], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha+1}, 2^{\alpha}3 \}$ if and only if $\alpha =1$. Furthermore, there exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha +1}, 2^{\alpha}3, 2^{\alpha +2}\}$ and $\alpha$ is odd if and only if $\alpha=1$. https://ijgt.ui.ac.ir/article_26288_3ae056425f24f8462629e53d162dedfc.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765012120230301New criteria for solvability, nilpotency and other properties of finite groups in terms of the order elements or subgroups35442628710.22108/ijgt.2022.131888.1766ENMarcelHerzogDipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), ItalyPatriziaLongobardiDipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), ItalyMercedeMajDipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), ItalyJournal Article20211215In this survey we shall describe some recent criteria for solvability, nilpotency and other properties of finite groups $G$, based either on the orders of the elements of $G$ or on the orders of the subgroups of $G$.https://ijgt.ui.ac.ir/article_26287_b289df06ad4eb960eea5c16bde5aa245.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765012120230301Existentially and $\kappa$-existentially closed groups45542635210.22108/ijgt.2022.131513.1758ENBurakKayaDepartment of Mathematics, Middle East Technical University, 06800, Ankara, TurkeyMahmutKuzucuoğluDepartment of Mathematics, Middle East Technical University, 06800,Ankara, TurkeyJournal Article20211117A group $G$ is existentially closed (algebraically closed) if every finite system of equations and in-equations that has coefficients in $G$ and has a solution in an overgroup $H\geq G$ has a solution in $G$. Existentially closed groups were introduced by W. R. Scott in 1951. B. H. Neumann posed the following question in 1973: Does there exist explicit examples of existentially closed groups? Generalized version of this question is as follows: Let $\kappa$ be an infinite cardinal. Does there exist explicit examples of $\kappa$-existentially closed groups? Recently an affirmative answer was given to Neumann's question and the generalized version of it, by Kaya-Kegel-Kuzucuo\u{g}lu. We give a survey of these results. We also prove that, there are maximal subgroups of $\kappa$-existentially existentially closed groups and provide some information about subgroups containing the centralizer of subgroups generated by fewer than $\kappa$-elements. This generalizes a result of Hickin-Macintyre.https://ijgt.ui.ac.ir/article_26352_bb5dd75b306e31d26a0a4627b820259e.pdf