International Journal of Group TheoryInternational Journal of Group Theory
http://ijgt.ui.ac.ir/
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http://ijgt.ui.ac.ir/
Feed provided by International Journal of Group Theory. Click to visit.On finite-by-nilpotent profinite groups
http://ijgt.ui.ac.ir/article_24082_0.html
Let $gamma_n=[x_1,dots,x_n]$ be the $n$th lower central word. Suppose that $G$ is a profinite groupwhere the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elementsfor any $x in G$. We prove that then $gamma_{n+1}(G)$ has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies thata profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements, for any $xin G$.Mon, 28 Oct 2019 20:30:00 +0100Proceedings of Ischia Group Theory 2018
http://ijgt.ui.ac.ir/article_24521_4453.html
Sun, 31 May 2020 19:30:00 +0100Catalan fragile words
http://ijgt.ui.ac.ir/article_23435_4453.html
‎Fragile words have been already considered in the context of automata groups‎. ‎Here we focus our attention on a special class of strongly fragile words that we call Catalan fragile words‎. ‎Among other properties‎, ‎we show that there exists a one-to-one correspondence between the set of Catalan fragile words and the set of full binary trees‎.Sun, 31 May 2020 19:30:00 +0100INFLUENCE OF COMPLEMENTED SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS
http://ijgt.ui.ac.ir/article_24261_0.html
P. Hall proved that a finite group G is supersoluble with elementary abelian Sylow sub-groups if and only if every subgroup of G is complemented in G. He called such groups complemented. A. Ballester-Bolinches and X. Guo established the structure of minimal non-complemented groups. We give the classification of finite non-soluble groups all of whose second maximal subgroups are complemented groups. We also prove that every finite group with less than 21 non-complemented non-minimal {2; 3; 5}--subgroups is soluble.Sun, 15 Dec 2019 20:30:00 +0100Topological loops with solvable multiplication groups of dimension at most six are centrally ...
http://ijgt.ui.ac.ir/article_23511_4453.html
The main result of our consideration is the proof of the centrally nilpotency of class two property for connected topological proper loops $L$ of dimension $le 3$ which have an at most six-dimensional solvable indecomposable Lie group as their multiplication group. This theorem is obtained from our previous classification by the investigation of six-dimensional indecomposable solvable multiplication Lie groups having a five-dimensional nilradical. We determine the Lie algebras of these multiplication groups and the subalgebras of the corresponding inner mapping groups.Sun, 31 May 2020 19:30:00 +0100Groups with numerical restrictions on minimal generating sets
http://ijgt.ui.ac.ir/article_23278_4453.html
We study an inverse problem of small doubling type. We investigate the structure of a finitely generated group $G$ such that, for any set $S$ of generators of $G$ of minimal order, we have $S^2 leq 3|S|-beta$, where $beta in {1, 2, 3}$Sun, 31 May 2020 19:30:00 +0100ENGEL GROUPS IN BATH - TEN YEARS LATER
http://ijgt.ui.ac.ir/article_24420_0.html
The eighth edition of the international series of Groups St Andrews conferences was held at the University of Bath in 2009 and one of the theme days was dedicated to Engel groups. Since then much attention has been devoted to a verbal generalization of Engel groups. In this paper we will survey the development of this investigation during the last decade.Sun, 26 Jan 2020 20:30:00 +0100A survey on groups with some restrictions on normalizers or centralizers
http://ijgt.ui.ac.ir/article_23436_4453.html
We consider conditions on normalizers or centralizers in a group and we collect results showing how such conditions influence the structure of the group.Sun, 31 May 2020 19:30:00 +0100The Fibonacci-Circulant Sequences in the Binary Polyhedral Groups
http://ijgt.ui.ac.ir/article_24424_0.html
Deveci et al. defined 6. the Fibonacci-circulant sequences of the first and second kinds as shown, respectively: x_{n}¹=-x_{n-1}¹+x_{n-2}¹-x_{n-3}¹ for n≥4, where x₁¹=x₂¹=0 and x₃¹=1and x_{n}²=-x_{n-3}²-x_{n-4}²+x_{n-5}² for n≥6, where x₁²=x₂²=x₃²=x₄²=0 and x₅²=1.Also, they extended the Fibonacci-circulant sequences of the first and second kinds to groups. In this work, we obtain the periods of the Fibonacci-circulant sequences of the first and second kinds in the binary polyhedral groups.Mon, 27 Jan 2020 20:30:00 +0100Finite groups with seminormal or abnormal Sylow subgroups
http://ijgt.ui.ac.ir/article_23213_0.html
Let $G$ be a finite group in which every Sylow subgroup is seminormal or abnormal. We prove that $G$ has a Sylow tower. We establish that if a group has a maximal subgroup with Sylow subgroups under the same conditions, then this group is soluble.Sun, 06 Jan 2019 20:30:00 +0100Open normal subgroups in normally constrained pro-$p$ groups
http://ijgt.ui.ac.ir/article_23588_4453.html
In this paper we analyse properties satisfied by certain open normal subgroups in normally constrained pro-‎$‎p‎$ groups and in a spread version of normally constrained pro-‎$‎p‎$‎ groups‎. ‎In the case of powerful normally constrained pro-‎$‎p‎$ groups‎, ‎we exhibit some kind of inheritance properties in certain open normal subgroups‎.Sun, 31 May 2020 19:30:00 +0100ALGORITHMIC PROBLEMS IN ENGEL GROUPS AND CRYPTOGRAPHIC APPLICATIONS
http://ijgt.ui.ac.ir/article_24453_0.html
The theory of Engel groups plays an important role in group theory since they are closely related to the Burnside problems. In this survey we consider several classical and novel algorithmic problemsfor Engel groups. We study these problems with a view towards applications to cryptography.Fri, 07 Feb 2020 20:30:00 +0100Faithful Real Representations of Groups of $F$-type
http://ijgt.ui.ac.ir/article_23755_0.html
‎Groups of $F$-type were introduced in [B‎. ‎Fine and G‎. ‎Rosenberger‎, ‎Generalizing Algebraic Properties of Fuchsian Groups‎, emph{London Math. Soc. Lecture Note Ser.}, textbf{159} (1991)‎ ‎124--147.] as a natural algebraic generalization of Fuchsian groups‎. ‎They can be considered as the analogs of cyclically pinched one-relator groups where torsion is allowed‎. ‎Using the methods In [B‎. ‎Fine‎. ‎M‎. ‎Kreuzer and G‎. ‎Rosenberger‎, ‎Faithful Real Representations of Cyclically Pinched One-Relator Groups, emph{Int. J. Group Theory}, textbf{3}‎ ‎(2014) 1--8.] we prove that any hyperbolic group of $F$-type has a faithful representation in $PSL(2,mathbb R)$‎. ‎From this we also obtain that a cyclically pinched one-relator group has a faithful real representation if and only if it is hyperbolic‎. ‎We further survey the many nice properties of groups of $F$-type‎. ‎Tue, 09 Jul 2019 19:30:00 +0100Integral forms in vertex operator algebras, a survey
http://ijgt.ui.ac.ir/article_24361_4453.html
We give a brief survey of recent work on integral forms in vertex operator algebras (VOAs).Sun, 31 May 2020 19:30:00 +0100Groups with many roots
http://ijgt.ui.ac.ir/article_24499_0.html
Given a prime $p$, a finite group $G$ and a non-identity element $g$, what is the largest number of $pth$ roots $g$ can have? We write $myro_p(G)$, or just $myro_p$, for the maximum value of $frac{1}{|G|}|{x in G: x^p=g}|$, where $g$ ranges over the non-identity elements of $G$. This paper studies groups for which $myro_p$ is large. If there is an element $g$ of $G$ with more $pth$ roots than the identity, then we show $myro_p(G) leq myro_p(P)$, where $P$ is any Sylow $p$-subgroup of $G$, meaning that we can often reduce to the case where $G$ is a $p$-group. We show that if $G$ is a regular $p$-group, then $myro_p(G) leq frac{1}{p}$, while if $G$ is a $p$-group of maximal class, then $myro_p(G) leq frac{1}{p} + frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $myro_2$, and give partial results on groups with high values of $myro_3$.Wed, 26 Feb 2020 20:30:00 +0100On Infinite Groups whose Finite Quotients have Restricted Prime Divisors
http://ijgt.ui.ac.ir/article_24518_0.html
The effect of restricting the set of primes dividing the orders of the finite quotients of a group is investigated. Particular attention is paid to abelian, soluble, locally soluble and locally finite groups. The connection with the extraction of roots is explored.Thu, 05 Mar 2020 20:30:00 +0100Some remarks on unipotent automorphisms
http://ijgt.ui.ac.ir/article_24519_0.html
In this paper we discuss some facts related to unipotent automorphisms of solvable groups.Thu, 05 Mar 2020 20:30:00 +0100On the power graphs of elementary abelian and extra special $p$-groups
http://ijgt.ui.ac.ir/article_24531_0.html
For a given odd prime $p$ we intend to study the power graphs of three classes of finite groups, the elementary abelian group of exponent $p$ and the extra special groups of exponents $p$ or $p^2$. We show that the power graphs are Eulerian, for every $p$. As a result of this study two classes of non-isomorphic groups have been specified where the power graphs are isomorphic. The Clique graphs of the power graphs of two considered classes have been proved to be complete as wellTue, 24 Mar 2020 19:30:00 +0100Small doubling in $m$-Engel groups
http://ijgt.ui.ac.ir/article_24532_0.html
We study some inverse problems of small doubling type in the class of $m$-Engel groups. In particular we investigate the structure of a finite subset $S$ of a torsion-free $m$-Engel group if $|S^2| = 2|S|+b$, where $0 leq b leq |S|-4$, for some values of $b$.Tue, 24 Mar 2020 19:30:00 +0100Omegas of agemos in powerful groups
http://ijgt.ui.ac.ir/article_23478_0.html
Let $G$ be a powerful $p$-group. In this note we investigate when the Omega subgroups of the Agemo subgroups of $G$ are Powerfully Nilpotent. We discuss the difference between the cases when $p=2$ and when $p$ is an odd prime. Moreover this work gives an example of a characteristic subgroup of $G^p$ which is Powerfully Nilpotent but not strongly powerful.Tue, 12 Mar 2019 20:30:00 +0100The character table of a sharply 5-transitive subgroup of the alternating group of degree 12
http://ijgt.ui.ac.ir/article_23524_0.html
In this paper we calculate the character table of a sharply $5$-transitive subgroup of ${rm Alt}(12)$, and of a sharply $4$-transitive subgroup of ${rm Alt}(11)$. Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.Fri, 12 Apr 2019 19:30:00 +0100Weakly totally permutable products and Fitting classes
http://ijgt.ui.ac.ir/article_23525_0.html
It is known that if $ G=AB $ is a product of its totally permutable subgroups $ A $ and $ B $‎, ‎then $ Gin mathfrak{F} $ if and only if $ Ain mathfrak{F} $ and $ Bin mathfrak{F} $ when $ mathfrak{F} $ is a Fischer class containing the class $ mathfrak{U} $ of supersoluble groups‎. ‎We show that this holds when $ G=AB $ is a weakly totally permutable product for a particular Fischer class‎, ‎$ mathfrak{F}diamond mathfrak{N} $‎, ‎where $ mathfrak{F} $ is a Fitting class containing the class $ mathfrak{U} $ and $ mathfrak{N} $ a class of nilpotent groups‎. ‎We also extend some results concerning the $ mathfrak{U} $-hypercentre of a totally permutable product to a weakly totally permutable product‎.Fri, 12 Apr 2019 19:30:00 +0100$4$-Regular prime graphs of nonsolvable groups
http://ijgt.ui.ac.ir/article_23718_0.html
Let $G$ be a finite group and $cd(G)$ denote the character degree set for $G$‎. ‎The prime graph $DG$ is a simple graph whose vertex set consists of prime divisors of elements in $cd(G)$‎, ‎denoted $rho(G)$‎. ‎Two primes $p,qin rho(G)$ are adjacent in $DG$ if and only if $pq|a$ for some $ain cd(G)$‎. ‎We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group‎.Mon, 17 Jun 2019 19:30:00 +0100Recognition of Janko groups and some simple $K_4$-groups by the order and one irreducible ...
http://ijgt.ui.ac.ir/article_23719_0.html
‎In this paper we prove that some Janko groups are uniquely‎ ‎determined by their orders and one irreducible character‎ ‎degree‎. ‎Also we prove that some finite simple $K_4$-groups are‎ ‎uniquely determined by their character degree graphs and their‎ ‎orders‎.Mon, 17 Jun 2019 19:30:00 +0100A note on locally soluble almost subnormal subgroups in divsion rings
http://ijgt.ui.ac.ir/article_23813_0.html
Let D be a division ring with center F and assume that N is a locally soluble almost subnormal subgroup of the multiplicative group D* of D. We prove that if N is algebraic over F, then N is central. This answers partially [11, Conjecture 1].Wed, 24 Jul 2019 19:30:00 +0100Hilbert's theorem 90 for finite nilpotent groups
http://ijgt.ui.ac.ir/article_23857_0.html
‎‎In this note we prove an analog of Hilbert's theorem 90 for finite nilpotent groups‎. ‎Our version of Hilbert's theorem 90 was inspired by the Boston--Bush--Hajir (BBH) heuristics in number theory and will be useful in extending the BBH heuristics beyond quadratic field extensions‎.Mon, 12 Aug 2019 19:30:00 +0100The minimum sum of element orders of finite groups
http://ijgt.ui.ac.ir/article_23858_0.html
‎Let $ G $ be a finite group and ( psi(G)=sum_{gin G}o(g) )‎, ‎where $ o(g) $ denotes the order of $gin G$‎. ‎We show that the Conjecture 4.6.5 posed in [Group Theory and Computation‎, ‎(2018) 59-90]‎, ‎is incorrect‎. ‎In fact‎, ‎we find a pair of finite groups $G$ and $S$ of the same order such that $ psi(G)<psi(S)$‎, ‎with $G$ solvable and $S$ simple‎.Tue, 13 Aug 2019 19:30:00 +0100Characterization of finite groups with a unique non-nilpotent proper subgroup
http://ijgt.ui.ac.ir/article_23859_0.html
‎‎We characterize finite non-nilpotent groups $G$ with a unique non-nilpotent proper subgroup‎. ‎We show that $|G|$ has at most‎ ‎three prime divisors‎. ‎When $G$ is supersolvable we find the presentation of $G$ and when $G$ is non-supersolvable we show that‎ ‎either $G$ is a direct product of an Schmidt group and a cyclic group or a semi direct product of a $p$-group by a cyclic group of prime power order‎.Tue, 13 Aug 2019 19:30:00 +0100Minimal embeddings of small finite groups
http://ijgt.ui.ac.ir/article_24034_0.html
We determine the groups of minimal order in which all groups of order $n$ can embedded for $ 1 le n le 15$‎. ‎We further determine the minimal order of a group in which all groups of order $n$ or less can be embedded‎, ‎also for $ 1 le n le 15$‎.Fri, 11 Oct 2019 20:30:00 +0100