International Journal of Group Theory
https://ijgt.ui.ac.ir/
International Journal of Group Theory
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Fri, 01 Dec 2023 00:00:00 +0330
Fri, 01 Dec 2023 00:00:00 +0330

Solvable groups whose monomial, monolithic characters have prime power codegrees
https://ijgt.ui.ac.ir/article_26289.html
In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$&nbsp;is a $p$power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi \in&nbsp;{\rm IBr} (G)$, then $P$ is normal in $G$, where $p$ is a prime and&nbsp;$P$ is a Sylow $p$subgroup of $G$.

Trilinear alternating forms and related CMLs and GECs
https://ijgt.ui.ac.ir/article_26353.html
The classification of trivectors(trilinear alternating forms) depends essentially on the dimension $n$ of the base space. This classification seems to be a difficult problem (unlike in the bilinear case). For $n\leq 8 $ there exist finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\bar{K}$(the algebraic closure of $K$) to $K.$ In this paper, we are interested in the classification of trivectors of an&nbsp;eight dimensional vector space over a finite field of characteristic $3,$ $% K=\mathbb{F}_{3^{m}}.$ We obtain a $31$ inequivalent trivectors, $20$ of which are full&nbsp;rank. Having its motivation in the theory of the generalized elliptic curves and commutative moufang loop, this research studies the case of the forms over the 3 elements field. We use a transfer theorem providing a onetoone correspondence between the classes of trilinear alternating forms of rank $8$ over a finite field with $3$ elements $\mathbb{F}_{3}$ and the rank $9$ class $2$ Hall generalized&nbsp;elliptic curves (GECs) of $3$order $9$ and commutative moufang loop (CMLs).&nbsp;We derive a classification and explicit descriptions of the $31$ Hall GECs&nbsp;whose rank and $3$order both equal $9$ and the number of order $3^{9}$CMLs.&nbsp;

Ramification structures for quotients of multiEGS groups
https://ijgt.ui.ac.ir/article_26749.html
Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely grouptheoretic condition, which is the existence of a socalled ramification structure. G&uuml;l and UriaAlbizuri showed that quotients of the periodic GrigorchukGuptaSidki groups, GGSgroups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGSgroups, namely multiEGS groups, also admit ramification structures.

Profinite just infinite residually solvable Lie algebras
https://ijgt.ui.ac.ir/article_26772.html
We provide some characterization theorems about just infinite profinite residually solvable Lie algebras, similarly to what C. Reid has done for just infinite profinite groups. In particular, we prove that a profinite residually solvable Lie algebra is just infinite if and only if its obliquity subalgebra has finite codimension in the Lie algebra, and we establish a criterion for a profinite residually solvable Lie algebra to be just infinite, looking at the finite Lie algebras occurring in the inverse system.

Remark on Laquer's theorem for circulant determinants
https://ijgt.ui.ac.ir/article_26698.html
Olga TausskyTodd suggested the problem of determining the possible values of integer circulant determinants.&nbsp;To solve a special case of the problem, Laquer gave a factorization of circulant determinants.&nbsp;In this paper,&nbsp;we give a modest generalization of Laquer's theorem.&nbsp;Also, we give an application of the generalization to integer group determinants.

Existence of rational primitive normal pairs over finite fields
https://ijgt.ui.ac.ir/article_26751.html
For a finite field $π½_{q^n}$ and a rational function $f=\frac{f_1}{f_2} \in π½_{q^n}(x)$, we present a sufficient condition for the existence of a primitive normal element $\alpha \in π½_{q^n}$ in such a way $f(\alpha)$ is also primitive in $π½_{q^n}$, where $f(x)$ is a rational function in $π½_{q^n}(x)$ of degree sum $m$ (degree sum of $f(x)=\frac{f_1(x)}{f_2(x)}$ is defined to be the sum of the degrees of $f_1(x)$ and $f_2(x)$). Additionally, for rational functions of degree sum 4, we proved that there are only $37$ and $16$ exceptional values of $(q,n)$ when $q=2^k$ and $q=3^k$ respectively.

Constructions and involutory properties in latin quandles
https://ijgt.ui.ac.ir/article_26773.html
This work studied involutory properties in Latin quandles using methods of quasiqroup theory, and classified latin quandle $Q$ into Left Involutory Property latin Quandle (LIPQ), Right Involutory Property Latin Quandle (RIPQ) and Involutory Property Latin Quandle (IPQ). It investigated a fourth property called the Cross Involutory Property Latin Quandle (CIPQ). The result showed that a latin quandle $Q$ that is a LIPQ and RIPQ is an IPQ. Moreover, it established that the necessary and sufficient conditions for a latin Alexander quandle $Q$ to be a CIPQ is that $b=t^2a +(1t)(tb+a)$ for all $a,b \in Q$ and $t\in A(Q)$.

Generalized order divisor graphs of finite group
https://ijgt.ui.ac.ir/article_27057.html
Let $G$ be a finite group and $k$ a fixed positive integer. We define the generalized order divisor graph of $G$ to be a graph whose vertex set is the group $G$ and in which two vertices $a$ and $b$ are adjacent if and only if the orders $o(a^k)$ and $o(b^k)$ are different and either $o(a^k)$ divides $o(b^k)$ or $o(b^k)$ divides $o(a^k)$. This generalizes the order divisor graphs of finite groups. Some properties of our graph are introduced, and we investigate the structure of the generalized order divisor graphs of finite cyclic groups.

On Neumannβs BFCtheorem and finitebynilpotent profinite groups
https://ijgt.ui.ac.ir/article_27090.html
Let $\gamma_{n}=[x_{1},\ldots,x_{n}]$ be the $n$th lower central word and $X_{n}(G)$ the set of $\gamma_{n}$values in a group $G$. Suppose that $G$ is a profinite group where, for each $g\in G$, there exists a positive integer $n=n(g)$ such that the set $g^{X_{n}(G)}=\{g^{y}\,\,y\in X_{n}(G)\}$ contains less than $2^{\aleph_{0}}$ elements. We prove that $G$ is a finitebynilpotent group.

Orbits classifying extensions of prime power order groups
https://ijgt.ui.ac.ir/article_27164.html
The strong isomorphism classes of extensions of finite groups are parametrized by orbits of a prescribed action on the second cohomology group. We study these orbits in the case of extensions of a finite abelian $p$group by a cyclic factor of order $p$. As an application, we compute the number and sizes of these orbits when the initial $p$group is generated by at most $3$ elements.

The influence of $\mathscr{H}$subgroups on $p$nilpotency and $p$supersolvability of finite groups
https://ijgt.ui.ac.ir/article_27183.html
Let $G$ be a finite group. A subgroup $H$ of $G$ is an $\mathscr{H}$subgroup in $G$ if $N_G(H)\cap H^g \leq H$ for any $g \in G$. In this article, by using the concept of $\mathscr{H}$subgroups, we study the influence of the intersection of $O^p(G_p^*)$ and the members of some fixed $\mathcal{M}_d(P)$ on the structure of the group $G$, where $P$ is a Sylow $p$subgroup of $G.$ Some new criteria for a group to be $p$nilpotent and $p$supersolvable are given and some recent results are extended and generalized.

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) 235265]. 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 $0\leq k\leq p1$, then this proportion is at most $(p\cdot k!)^{1}$ with equality if and only if $p\leq n&lt;2n$.

Noninner automorphisms of order $p$ in finite $p$groups of coclass $4$ and $5$
https://ijgt.ui.ac.ir/article_27436.html
A longstanding conjecture asserts that every finite nonabelian $p$group has a noninner automorphism of order $p$. This paper proves the conjecture for finite $p$groups of coclass $4$ and $5$ ($p\ge 5$). We also prove the conjecture for an odd order nonabelian $p$group $G$ with cyclic center satisfying $C_G(G^p\gamma_3(G))\cap Z_3(G)\le Z(\Phi(G))$.

Orders of simple groups and the BatemanHorn Conjecture
https://ijgt.ui.ac.ir/article_27521.html
We use the BatemanHorn 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 $p^21$ 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 $n\ge 5$.

An example of a quasicommutative inverse semigroup
https://ijgt.ui.ac.ir/article_27531.html
Constructing concrete examples of certain semigroups could help in implementing algorithms optimized for the users. We give concrete examples of certain finitely presented semigroups, namely $S_{p,n}$. Both computational and theoretical approaches are used for studying their structural properties to show that they are quasicommutative and inverse semigroups.

GowTamburini type generation of the special linear group for some special rings.
https://ijgt.ui.ac.ir/article_27604.html
Let $R$ be a commutative ring with unity and let $n\geq 3$ be an integer. Let $SL_n(R)$ and $E_n(R)$ denote respectively the special linear group and elementary subgroup of the general linear group $GL_n(R).$ A result of Hurwitz says that the special linear group of size atleast three over the ring of integers of an algebraic number field is finitely generated. A celebrated theorem in group theory states that finite simple groups are twogenerated. Since the special linear group of size atleast three over the ring of integers is not a finite simple group, we expect that it has more than two generators. In the special case, where $R$ is the ring of integers of an algebraic number field which is not totally imaginary, we provide for $E_n(R)$ (and hence $SL_n(R)$) a set of GowTamburini matrix generators, depending on the minimal number of generators of $R$ as a $Z$module.

Posetblowdowns of generalized quaternion groups
https://ijgt.ui.ac.ir/article_27671.html
Posetblowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Posetblowdowns are classified into three types: \textit{tame, wild}, and \textit{hybrid} depending on the sizes of their fibers. In this paper we describe the posetblowdowns for generalized quaternion groups $Q_{2^n}$ $(n \geq 3)$. They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive posetblowdowns associated with the three chief series of $Q_{2^n}$.

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 $64$.

Cubic semisymmetric graphs of order $44p$ or $44p^{2}$
https://ijgt.ui.ac.ir/article_27739.html
A simple graph is called semisymmetric if it is regular and edgetransitive but not vertextransitive. Let $p$ be an arbitrary prime. Folkman [J. Folkman, Regular linesymmetric graphs, J. Combinatorial Theory, \textbf{3} (1967) 215232.] proved that there are no cubic semisymmetric graphs of order $2p$ or $2p^{2}$. In this paper, an extension of his result in the case of cubic graphs of order $44p$ or $44p^{2}$ is given. By using group theoretic methods, we prove that there are no connected cubic semisymmetric graphs of order $44p$ or $44p^{2}$.

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 ``cSubnormal 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.

Nonseparable 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 nonseparable 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 nonseparable matrices of required types are derived. These can retain properties of lower order matrices or have new desirable properties. Infinite series of required type nonseparable matrices are constructible by the general methods.&nbsp;Nonseparable 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 nonseparable 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.

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 $G$.

The probability of zero multiplication in finite group algebras
https://ijgt.ui.ac.ir/article_27800.html
Let $\mathbb{F}_qG$ be a finite group algebra. We denote by $P(\mathbb{F}_qG)$ the probability that the product of two elements of $\mathbb{F}_qG$ be zero. In this paper, we obtain several results on this probability including a computing formula and characterizations. In particular, the computing formula for the $P(\mathbb{F}_qG)$ are established where $G$ is the cyclic group $C_n$, the Quaternion group $Q_8$, the symmetric group $S_3$ and $F_q$ is a finite field.

Structure of finite groups with trait of nonnormal subgroups II
https://ijgt.ui.ac.ir/article_27804.html
A finite nonDedekind group $G$ is called an π©ππgroup if all nonnormal abelian subgroups are cyclic. In this paper, all finite π©ππgroups will be characterized. Also, it will be shown that the center of nonnilpotent π©ππ groups is cyclic. If π©ππgroup $G$ has a nonabelian nonnormal Sylow subgroup of odd order, then other Sylow subgroups of $G$ are cyclic or of quaternion type.

Average order in regular wreath products
https://ijgt.ui.ac.ir/article_27816.html
We obtain an exact formula for the average order of elements of regular wreath product of two finite groups. Then focussing our attention on $p$groups for primes $p$, we give an estimate for the verage order of a wreath product $A\wr B$ in terms of maximum order of elements of $A$ and average order of $B$ and an exact formula for the distribution of orders of elements of $A\wr B.$ Finally, we show how wreath products can be used to find several rational numbers which are limits of average orders of a sequence of $p$groups with cardinalities going to infinity.

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.

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.

Groups having $11$ cyclic subgroups
https://ijgt.ui.ac.ir/article_27858.html
Let $c(G)$ denotes the number of cyclic subgroups of a finite group $G$. A group $G$ is said to be {\em $n$cyclic}, if $c(G)=n$. In this paper, we classify all $11$cyclic groups.

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 nonascendant subgroups in term of chain conditions and of ascendant subgroups.