University of IsfahanInternational Journal of Group Theory2251-765013320240901Preface of the 2022 CCGTA IN SOUTH FLA28322ENJournal Article20240406https://ijgt.ui.ac.ir/article_28322_a6c211d4b99ffe712d2141bfd2a72e22.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765013320240901New constructions of Deza digraphs2252402730110.22108/ijgt.2023.136268.1823ENDeanCrnkovicFaculty of Mathematics, University of Rijeka, Rijeka, Croatia0000-0002-3299-7859HadiKharaghaniDepartment of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Canada0000-0003-0115-7117ShoSudaDepartment of Mathematics, National Defense Academy of Japan, 2-10-20 Hashirimizu, Yokosuka, Kanagawa, Japan0000-0002-5280-2668AndreaSvobFaculty of Mathematics, University of Rijeka, Rijeka, Croatia0000-0001-6558-5167Journal Article20221230Deza 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.https://ijgt.ui.ac.ir/article_27301_a9626ce302ed5d65efb57be7cdcc69c9.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765013320240901Computing Galois groups2412502731510.22108/ijgt.2023.136401.1825ENAndreas-StephanElsenhansDepartment of Mathematics, University of Würzburg, Würzburg, Germany0000-0003-2741-998XJournal Article20230118The 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 <em>magma</em> [W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, <em>J. Symbolic Comput.</em>, <strong>24</strong> (1997) 235--265]. implementation.https://ijgt.ui.ac.ir/article_27315_313de2fc1cff1c4f424d267af7a802d9.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765013320240901On the proportion of elements of prime order in finite symmetric groups2512562743510.22108/ijgt.2023.135509.1810ENCheryl E.PraegerCentre for the Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Perth
6009, Australia0000-0002-0881-7336EnochSuleimanDepartment of Mathematics, Federal University Gashua, Yobe State, NigeriaJournal Article20221025We 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 $p\leq n<2n$.https://ijgt.ui.ac.ir/article_27435_d4e69a5b8775bbd89c1f00b89aa16ddf.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765013320240901Orders of simple groups and the Bateman--Horn Conjecture2572692752110.22108/ijgt.2023.136666.1828ENGareth AneurinJonesDepartment of Mathematics, School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UKAlexander K.ZvonkinLaBRI, Université de Bordeaux, 351 Cours de la Libération, F-33405, Talence, FranceJournal Article20230208We 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 $n\ge 5$.https://ijgt.ui.ac.ir/article_27521_b123f0c513f61f9c023d31a0e1bfa7c8.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765013320240901Non-separable matrix builders for signal processing, quantum information and mimo applications2712912776410.22108/ijgt.2023.136437.1824ENTedHurleyUniversity of Galway (Formerly National University of Ireland Galway), IrelandBarryHurleyFriar's Hill, Galway, IrelandJournal Article20230113Matrices 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 <em>separable </em>matrix. A <em>non-separable</em> matrix is a matrix which is not separable and is often referred to as <em>an entangled matrix</em>. 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.<br /> <br />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 <em>multidimensional entangled paraunitary matrices</em>; 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.https://ijgt.ui.ac.ir/article_27764_19d93a040e483ca055e20957ac19deb1.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765013320240901Covering perfect hash families and covering arrays of higher index2933052785710.22108/ijgt.2023.137230.1836ENCharles J.ColbournComputing and Augmented Intelligence, Arizona State University, PO Box 878809, Tempe, AZ, 85287-8809, U.S.A.0000-0002-3104-9515Journal Article20230331By 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 $\lambda > 1$ 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.https://ijgt.ui.ac.ir/article_27857_35a8cbe8f0983688226c153eb2bd6d10.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765013320240901Enumerating word maps in finite groups3073182799610.22108/ijgt.2023.136972.1833ENBogdan S.ChlebusSchool of Computer and Cyber Sciences, Augusta University, Augusta, GA, USA0000-0003-4884-941XWilliamCockeSchool of Computer and Cyber Sciences, Augusta University, Augusta, GA, USA0000-0002-0732-6666Meng-CheHoDepartment of Mathematics, California State University, Northridge, Northridge, CA, USA0000-0001-6292-6835Journal Article20230301We 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.<br /> <br />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.https://ijgt.ui.ac.ir/article_27996_bc71bfef6ef95f19a0ff7d04f082cbc5.pdf