International Journal of Group Theory
https://ijgt.ui.ac.ir/
International Journal of Group Theoryendaily1Fri, 01 Dec 2023 00:00:00 +0330Fri, 01 Dec 2023 00:00:00 +0330Solvable 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 one-to-one 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 multi-EGS 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 group-theoretic condition, which is the existence of a so-called ramification structure. G&uuml;l and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups, namely multi-EGS 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 Taussky-Todd 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 $š¯”½_{q^n}$ and a rational function $f=\frac{f_1}{f_2} \in š¯”½_{q^n}(x)$, we present a sufficient condition for the existence of a primitive normal element $\alpha \in š¯”½_{q^n}$ in such a way $f(\alpha)$ is also primitive in $š¯”½_{q^n}$, where $f(x)$ is a rational function in $š¯”½_{q^n}(x)$ 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 $b=t^2a +(1-t)(tb+a)$ 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 Neumannā€™s BFC-theorem and finite-by-nilpotent 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 finite-by-nilpotent 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) 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 $p\leq n&lt;2n$.Non-inner automorphisms of order $p$ in finite $p$-groups of coclass $4$ and $5$
https://ijgt.ui.ac.ir/article_27436.html
A long-standing conjecture asserts that every finite nonabelian $p$-group has a non-inner 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 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 $n\ge 5$.An example of a quasi-commutative 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 quasi-commutative and inverse semigroups.Gow-Tamburini 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 two-generated. 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 Gow-Tamburini matrix generators, depending on the minimal number of generators of $R$ as a $Z$-module.