International Journal of Group Theory
https://ijgt.ui.ac.ir/
International Journal of Group Theoryendaily1Sat, 16 Dec 2023 00:00:00 +0330Sat, 16 Dec 2023 00:00:00 +0330On the structure of some left braces
https://ijgt.ui.ac.ir/article_27997.html
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.An overview of torus fully homomorphic encryption
https://ijgt.ui.ac.ir/article_28010.html
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, STOC '09: Proceedings of the forty-first annual ACM symposium on Theory of computing, 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.A generalization of the Chermak--Delgado measure on subgroups and its associated lattice
https://ijgt.ui.ac.ir/article_28020.html
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.On numbers which are orders of nilpotent groups with bounded class
https://ijgt.ui.ac.ir/article_28077.html
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 $n$.On some groups whose subnormal subgroups are contranormal-free
https://ijgt.ui.ac.ir/article_28378.html
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.A characterization of $A_5$ by its average order
https://ijgt.ui.ac.ir/article_28078.html
Let $o(G)$ be the average order of a finite group $G$. M. Herzog, P. Longobardi and M. Maj [M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, 597 (2022) 1-23.] showed that if $G$ is non-solvable and $o(G)=o(A_5)$, then $G\cong A_5$. In this note, we prove that the equality $o(G)=o(A_5)$ does not hold for any finite solvable group $G$. Consequently, up to isomorphism,$A_5$ is determined by its average order.Cartesian symmetry classes associated with certain subgroups of $S_m$
https://ijgt.ui.ac.ir/article_28294.html
In this paper, the problem existing $O$-basis for Cartesian symmetry classes is discussed. The dimensions of Cartesian symmetry classes associated with a cyclic subgroup of the symmetric group $S_m$ (generated by a product of disjoint cycles) and the product of cyclic subgroups of $S_m$ are explicitly expressed in terms of the Ramanajun sum. Additionally, a necessary and sufficient condition for the existence of an $O$-basis for Cartesian symmetry classes associated with the irreducible characters of dihedral group is given. The dimensions of these classes are also computed.On a question of Jaikin-Zapirain about the average order elements of finite groups
https://ijgt.ui.ac.ir/article_28548.html
For a finite group $G$, the average order $o(G)$ is defined to be the average of all order elements in $G$, that is $o( G)=\frac{1}{|G|}\sum_{x\in G}o(x)$, where $o(x)$ is the order of element $x$ in $G$. Jaikin-Zapirain in [On the number of conjugacy classes of finite nilpotent groups, Advances in Mathematics, \textbf{227} (2011) 1129-1143] asked the following question: if $G$ is a finite ($p$-) group and $N$ is a normal (abelian) subgroup of $G$, is it true that $o(N)^{\frac{1}{2}}\leq o(G) $? We say that $G$ satisfies the average condition if $o(H)\leq o(G)$, for all subgroups $H$ of $G$. In this paer we show that every finite abelian group satisfies the average condition. This result confirms and improves the question of Jaikin-Zapirain for finite abelian groups.Quantitative characterization of finite simple groups: a complement
https://ijgt.ui.ac.ir/article_28685.html
In this paper, we summarize the work on the characterization of finite simple groups and the study on finite groups with ``the set of element orders&quot; and ``two orders&quot; (the order of group and the set of element orders). Some related topics, and the applications together with their generalizations are also discussed. The original version of this article was published in Chinese in the journal Scientia Sinica Mathematica, no.53(2023), pp.931-952. This revised and expanded version has corrected several errors and added quite a few contents. Especially, it is pointed that this work has applications in mathematics and computational complexity theory.
In this paper, we summarize the work on the characterization of finite simple groups and the study on finite groups with ``the set of element orders&quot; and ``two orders&quot; (the order of group and the set of element orders). Some related topics, and the applications together with their generalizations are also discussed. The original version of this article was published in Chinese in the journal Scientia Sinica Mathematica, no.53(2023), pp.931-952. This revised and expanded version has corrected several errors and added quite a few contents. Especially, it is pointed that this work has applications in mathematics and computational complexity theory.Representation theory of skew braces
https://ijgt.ui.ac.ir/article_28692.html
Skew braces are generalizations of groups in a sense. Hence the representations of skew braces should be considered. According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility condition. Following their definition, we introduce the concepts of modules over skew braces, homomorphisms, isomorphism, indecomposable modules, simple modules, semisimple modules. In addition, we define trivial modules, tensor modules, simple modules and regular modules for skew braces. We shall explain how some of the results from representation theory of groups, such as Maschke&rsquo;s theorem and its inverse and Clifford&rsquo;s theorem, extend naturally to that of skew braces. We shall also give some concrete examples for some kinds of skew braces to illustrate that skew brace representations are more difficult to classify than group representations.