University of Isfahan International Journal of Group Theory 2251-7650 14 2 2023 12 16 On the structure of some left braces 47 58 27997 10.22108/ijgt.2023.139145.1872 EN Adolfo Ballester-Bolinches Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100, Burjassot, València, Spain Ramón Esteban-Romero Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100, Burjassot, València, Spain Leonid A. Kurdachenko Department of Geometry and Algebra, Oles Honchar Dnipro National University Departament de Matemàtiques, Universitat de València Vicent Pérez-Calabuig Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100, Burjassot, València, Spain Journal Article 2023 09 15 Given an element $a$ of a left brace $A$ satisfying some nilpotency conditions, we describe the smallest subbrace of $A$ containing~$a$. We also present a description of the left braces satisfying the minimal condition for subbraces.

https://ijgt.ui.ac.ir/article_27997_8c6a7824288ee189dc5ba258e549c054.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 2 2023 12 29 An overview of torus fully homomorphic encryption 59 73 28010 10.22108/ijgt.2023.139030.1869 EN Maria Ferrara Dipartimento di Matematica e Fisica, Università della Campania “Luigi Vanvitelli”, viale Lincoln, 5 - 81100, Italy Antonio Tortora Dipartimento di Matematica e Fisica, Università della Campania “Luigi Vanvitelli”, viale Lincoln, 5 - 81100, Italy Maria Tota Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132 - 84084 - Fisciano (SA), Italy Journal Article 2023 09 05 The homomorphic encryption allows us to operate on encrypted data, making any action less vulnerable to hacking. The implementation of a fully homomorphic cryptosystem has long been impracticable. A breakthrough was achieved only in 2009 thanks to Gentry [C. Gentry, Fully homomorphic encryption using ideal lattices,<em> STOC '09: Proceedings of the forty-first annual ACM symposium on Theory of computing</em>, Association for Computing Machinery, New York, (2009) 169--178.] with his innovative idea of bootstrapping. TFHE is a torus-based fully homomorphic cryptosystem using the bootstrapping technique. This paper aims to present TFHE from an algebraic point of view.

https://ijgt.ui.ac.ir/article_28010_ce3bd36fdffff4ea67a1dd4de2776590.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 2 2023 12 16 A generalization of the Chermak--Delgado measure on subgroups and its associated lattice 75 91 28020 10.22108/ijgt.2023.138469.1859 EN William Cocke School of Computer and Cyber Sciences, Augusta University, 100 Grace Hopper Ln, Augusta, GA 30901, USA Luise-Charlotte Kappe Department of Mathematics and Statistics, Binghamton University, P.O. Box 6000, Binghamton, NY 13902-6000, USA Arturo Magidin Department of Mathematics, University of Louisiana at Lafayette, P.O. Box 43568, Lafayette LA 70504-3568 USA Journal Article 2023 07 20 We generalize the Chermak--Delgado measure of a subgroup of a finite group $G$, $\mu(H) = |H||C_{G}(H)|$, and its associated lattice of subgroups with maximal measure. We consider mappings $M$ of the lattice of all subgroups $\mathrm{Sub}(G)$ into itself and define a measure associated to $M$ by setting $\mu(H)=|H||M(H)|$. We investigate under what conditions on $M$ the subgroups with maximal measure form a sublattice of $\mathrm{Sub}(G)$. In particular, our focus is on the case where $M(H)$ is a centralizer-like subgroup.

https://ijgt.ui.ac.ir/article_28020_2cf9977be55d3a3fb6b0299966dee9c7.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 2 2024 01 06 On numbers which are orders of nilpotent groups with bounded class 93 97 28077 10.22108/ijgt.2023.137950.1851 EN Maria Ferrara Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, viale Lincoln 5, Caserta (Italy) Journal Article 2023 06 06 Let $n$ be a positive integer. In this short note, we characterize those numbers $m$ for which any group of order $m$ is an $n$-Engel group and those numbers $m$ for which any group of order $m$ has all its subgroups subnormal of defect at most $n$.

https://ijgt.ui.ac.ir/article_28077_e4ab13352bc7d2c98448c6417cf510f7.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 2 2024 05 05 On some groups whose subnormal subgroups are contranormal-free 99 115 28378 10.22108/ijgt.2024.139136.1871 EN Leonid A. Kurdachenko Department of Algebra and Geometry, School of Mathematics and Mechanics, National Dnipro University, Gagarin Prospect 72, Dnipro 10, 49010 Ukraine Patrizia Longobardi Department of Mathematics, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy Mercede Maj Department of Mathematics, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy Journal Article 2023 09 15 If $G$ is a group, a subgroup $H$ of $G$ is said to be contranormal in $G$ if $H^G = G$, where $H^G$ is the normal closure of $H$ in $G$. We say that a group is contranormal-free if it does not contain proper contranormal subgroups. Obviously, a nilpotent group is contranormal-free. Conversely, if $G$ is a finite contranormal-free group, then $G$ is nilpotent. We study (infinite) groups whose subnormal subgroups are contranormal-free. We prove that if $G$ is a group which contains a normal nilpotent subgroup $A$ such that $G/A$ is a periodic Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is generated by subnormal nilpotent subgroups; in particular $G$ is a Baer group. Furthermore, if $G$ is a group which contains a normal nilpotent subgroup $A$ such that the $0$-rank of $A$ is finite, the set $\Pi(A)$ is finite, $G/A$ is a Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is a Baer group.

https://ijgt.ui.ac.ir/article_28378_baba939854ad1ebe4110fdb3b3f960d7.pdf