International Journal of Group Theory
https://ijgt.ui.ac.ir/
International Journal of Group Theoryendaily1Sun, 01 Sep 2024 00:00:00 +0330Sun, 01 Sep 2024 00:00:00 +0330Preface of the 2022 CCGTA IN SOUTH FLA
https://ijgt.ui.ac.ir/article_28322.html
New constructions of Deza digraphs
https://ijgt.ui.ac.ir/article_27301.html
Deza digraphs were introduced in 2003 by Zhang and Wang as directed graph version of Deza graphs, that also generalize the notion of directed strongly regular graphs. In this paper, we give several new constructions of Deza digraphs. Further, we introduce twin and Siamese twin (directed) Deza graphs and construct several examples. Finally, we study a variation of directed Deza graphs and provide a construction from finite fields.Computing Galois groups
https://ijgt.ui.ac.ir/article_27315.html
The determination of a Galois group is an important question in computational algebraic number theory. One approach is based on the inspection of resolvents. This article reports on this method and on the performance of the current magma [W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997) 235--265]. implementation.On the proportion of elements of prime order in finite symmetric groups
https://ijgt.ui.ac.ir/article_27435.html
We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$ with $0\leq k\leq p-1$, then this proportion is at most $(p\cdot k!)^{-1}$ with equality if and only if $p\leq n&lt;2n$.Orders of simple groups and the Bateman--Horn Conjecture
https://ijgt.ui.ac.ir/article_27521.html
We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and H\"older in the 1890s.) The groups satisfying this condition are ${\rm PSL}_2(8)$, ${\rm PSL}_2(9)$ and ${\rm PSL}_2(p)$ for primes $p$ such that $p^2-1$ is a product of six primes. The conjecture suggests that there are infinitely many such primes $p$, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, the construction of highly symmetric block designs, and the possible existence of infinitely many K$n$ groups for each $n\ge 5$.Non-separable matrix builders for signal processing, quantum information and mimo applications
https://ijgt.ui.ac.ir/article_27764.html
Matrices are built and designed by applying procedures from lower order matrices. Matrix tensor products, direct sums or multiplication of matrices are such procedures and a matrix built from these is said to be a&nbsp; separable matrix. A non-separable matrix is a matrix which is not separable and is often referred to as an entangled matrix. The matrices built may retain properties of the lower order matrices or may also acquire new desired properties not inherent in the constituents. Here design methods for non-separable matrices of required types are derived. These can retain properties of lower order matrices or have new desirable properties. Infinite series of required type non-separable matrices are constructible by the general methods.&nbsp;Non-separable matrices of required types are required for applications and other uses; they can capture the structure in a unique way and thus perform much better than separable matrices. General new methods are developed with which to construct multidimensional entangled paraunitary matrices; these have applications for wavelet and filter bank design. The constructions are used to design new systems of non-separable unitary matrices; these have applications in quantum information theory. Some consequences include the design of full diversity constellations of unitary matrices, which are used in MIMO systems, and methods to design infinite series of special types of Hadamard matrices.Covering perfect hash families and covering arrays of higher index
https://ijgt.ui.ac.ir/article_27857.html
By exploiting symmetries of finite fields, covering perfect hash families provide a succinct representation for covering arrays of index one. For certain parameters, this connection has led to both the best current asymptotic existence results and the best known efficient construction algorithms for covering arrays. The connection generalizes in a straightforward manner to arrays in which every $t$-way interaction is covered $\lambda &gt; 1$ times, i.e., to covering arrays of index more than one. Using this framework, we focus on easily computed, explicit upper bounds on numbers of rows for various parameters with higher index.Enumerating word maps in finite groups
https://ijgt.ui.ac.ir/article_27996.html
We consider word maps over finite groups. An $n$-variable word $w$ is an element of the free group on $n$-symbols. For any group $G$, a word $w$ induces a map from $G^n\mapsto G$ where $(g_1,\ldots,g_n)\mapsto w(g_1,\ldots,g_n)$. We observe that many groups have word maps that decompose into components. Such a decomposition facilitates a recursive approach to studying word maps. Building on this observation, and combining it with relevant properties of the word maps, allows us to develop an algorithm to calculate representatives of all the word maps over a finite group. Given these representatives, we can calculate word maps with specific properties over a given group, or show that such maps do not exist. In particular, we have computed an explicit a word on $A_5$ such that only generating tuples are nontrivial in its image.&nbsp;We also discuss how our algorithm could be used to computationally address many open questions about word maps. Promising directions of potential applications include Amit's conjecture, questions of chirality and rationality, and the search for multilinear maps over a group. We conclude with open questions regarding these problems.Determination of the left braces of order $64$
https://ijgt.ui.ac.ir/article_27709.html
We review some of the algorithms used for the determination of the left braces of order $64$.A study on the structure of finite groups with $c-$ subnormal subgroups
https://ijgt.ui.ac.ir/article_27757.html
In this paper, we use the definition of the concept ``c-Subnormal Subgroup" to study the structure of a given finite group $G$ which contains some $c-$subnormal subgroups. We prove two main theorems, which answer the question of what conditions must hold so that $G$ is an element in a formation $ \mathfrak{U}$ of supersoluble groups. Finally, we state many previous results which can be considered as special cases of these theorems.A lattice theoretic characterization for the existence of a faithful irreducible representation
https://ijgt.ui.ac.ir/article_27765.html
In a recent article S\'ebastien Palcoux formulated a sufficient condition on the subgroup lattice of a finite group $G$ that guarantees the existence of a faithful irreducible complex representation of $G$, and asked whether his condition is also necessary. In this short note we give an affirmative answer using Kochend\"orffer's criterion for the existence of a faithful irreducible representation based on the structure of the socle of $G$.Group nilpotency from a graph point of view
https://ijgt.ui.ac.ir/article_27840.html
Let $\Gamma_G$ denote a graph associated with a group $G$. A compelling question about finite groups asks whether or not a finite group $H$ must be nilpotent provided $\Gamma_H$ is isomorphic to $\Gamma_G$ for a finite nilpotent group $G$. In the present work we analyze the problem for different graphs that one can associate with a finite group, both reporting on existing answers and contributing to new ones.On groups satisfying the double chain condition on nonascendant subgroups
https://ijgt.ui.ac.ir/article_27867.html
If $\theta$ is a subgroup property, a group $G$ is said to satisfy the double chain condition on $\theta$-subgroups if it admits no infinite double chain $$\cdots&lt;X_{-n}&lt;\cdots&lt;X_{-1}&lt;X_0&lt;X_1&lt;\cdots&lt;X_n&lt;\cdots$$ consisting of $\theta$-subgroups. Here we want to describe the structure of locally finite and locally nilpotent groups satisfying the double chain condition on non-ascendant subgroups in term of chain conditions and of ascendant subgroups.Groups with the real chain condition on non-pronormal subgroups
https://ijgt.ui.ac.ir/article_27920.html
It is shown that a generalised radical group has no chain of non-pronormal subgroups with the same order type as the set $\mathbb{R}$ of the real numbers if and only if either the group is minimax or all subgroups are pronormal.On groups with many normal subgroups
https://ijgt.ui.ac.ir/article_27933.html
The structure of groups which are rich in normal subgroups has been investigated by several authors. Here, we prove that if a radical group G has normal deviation, which means that the set of its non-normal subgroups satisfies a very weak chain condition, then either G is a minimax group or all its subgroups are normal.Generalized nilpotent braces and nilpotent groups
https://ijgt.ui.ac.ir/article_27989.html
The authors give a brief survey of some results concerning nilpotent braces and their generalizations. Various results concerning $\star$-hypercentral and locally $\star$-nilpotent braces are given.On 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 $n$.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 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.