International Journal of Group TheoryInternational Journal of Group Theory
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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 +0100A probabilistic version of a theorem of László Kovács and Hyo-Seob Sim
http://ijgt.ui.ac.ir/article_23073_0.html
For a finite group group‎, ‎denote by $mathcal V(G)$ the smallest positive integer $k$ with the property that the probability of generating $G$ by $k$ randomly chosen elements is at least $1/e.$ Let $G$ be a finite soluble group‎. ‎{Assume} that for every $pin pi(G)$ there exists $G_pleq G$ such that $p$ does not divide $|G:G_p|$ and ${mathcal V}(G_p)leq d.$ Then ${mathcal V}(G)leq d+7.$‎Mon, 19 Nov 2018 20:30:00 +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 +0100Integral forms in vertex operator algebras, a survey
http://ijgt.ui.ac.ir/article_24361_0.html
We give a brief survey of recent work on integral forms in vertex operator algebras (VOAs).Tue, 07 Jan 2020 20:30:00 +0100On some generalization of the malnormal subgroups
http://ijgt.ui.ac.ir/article_23126_0.html
‎A subgroup $H$ of a group $G$ is called malonormal in $G$ if $H cap H^x =‎‎langle 1rangle$ for every element $x notin N_G(H)$‎. ‎These subgroups are‎ ‎generalizations of malnormal subgroups‎. ‎Every malnormal subgroup is‎ ‎malonormal‎, ‎and every selfnormalizing malonormal subgroup is malnormal‎. ‎Furthermore‎, ‎every normal subgroup is malonormal‎. ‎In this paper we obtain a‎ ‎description of finite and certain infinite groups‎, ‎whose subgroups are‎ ‎malonormal‎.Sat, 08 Dec 2018 20:30:00 +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 +0100$4$-quasinormal subgroups of prime order
http://ijgt.ui.ac.ir/article_23127_0.html
‎‎Generalizing the concept of quasinormality‎, ‎a subgroup $H$ of a group $G$ is said to be 4-quasinormal in $G$ if‎, ‎for all cyclic subgroups $K$ of $G$‎, ‎$langle H,Krangle=HKHK$‎. ‎An intermediate concept would be 3-quasinormality‎, ‎but in finite $p$-groups‎ - ‎our main concern‎ - ‎this is equivalent to quasinormality‎. ‎Quasinormal subgroups have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal subgroups‎, ‎particularly in finite‎ ‎$p$-groups‎. ‎However‎, ‎even in the smallest case‎, ‎when $H$ is a 4-quasinormal subgroup of order $p$ in a finite $p$-group $G$‎, ‎precisely how $H$ is embedded in $G$‎ ‎is not immediately obvious‎. ‎Here we consider one of these questions regarding the commutator subgroup $[H,G]$‎.Sat, 08 Dec 2018 20:30:00 +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 +0100The number of maximal subgroups and probabilistic generation of finite groups
http://ijgt.ui.ac.ir/article_23260_0.html
In this survey we present some significant bounds for the number of maximal subgroups of a given index of a finite group. As a consequence, new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability which are significantly tighter than the ones obtained in the paper of Jaikin-Zapirain and Pyber (Random generation of finite and profinite groups and group enumeration, Ann. Math., 183:769--814, 2011) are obtained. The results of Jaikin-Zapirain and Pyber, as well as other results of Lubotzky, Detomi, and Lucchini, appear as particular cases of our theorems.Wed, 23 Jan 2019 20:30:00 +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 +0100Groups with many self-centralizing or self-normalizing subgroups
http://ijgt.ui.ac.ir/article_23277_0.html
The purpose of this paper is to present a comprehensive overview of known and new results concerning the structure of groups in which all subgroups‎, ‎except those having a given property‎, ‎are either self-centralizing or self-normalizing‎.Thu, 31 Jan 2019 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 +0100Groups with numerical restrictions on minimal generating sets
http://ijgt.ui.ac.ir/article_23278_0.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 $beta in {1, 2, 3}$Thu, 31 Jan 2019 20:30:00 +0100Some problems about products of conjugacy classes in finite groups
http://ijgt.ui.ac.ir/article_23434_0.html
‎We summarize several results about non-simplicity‎, ‎solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes‎. ‎We also collect some problems that have only been partially solved‎.Wed, 06 Mar 2019 20:30:00 +0100Catalan fragile words
http://ijgt.ui.ac.ir/article_23435_0.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.Wed, 06 Mar 2019 20:30:00 +0100A survey on groups with some restrictions on normalizers or centralizers
http://ijgt.ui.ac.ir/article_23436_0.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.Wed, 06 Mar 2019 20:30:00 +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 +0100Topological loops with solvable multiplication groups of dimension at most six are centrally ...
http://ijgt.ui.ac.ir/article_23511_0.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.Wed, 27 Mar 2019 19:30:00 +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} $-hypercentre of a totally permutable product to a weakly totally permutable product‎.Fri, 12 Apr 2019 19:30:00 +0100Open normal subgroups in normally constrained pro-$p$ groups
http://ijgt.ui.ac.ir/article_23588_0.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.Wed, 01 May 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 one-prime power hypothesis for conjugacy classes restricted to normal subgroups and ...
http://ijgt.ui.ac.ir/article_23001_4261.html
We say that a group $G$ satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers‎. ‎Insoluble groups which satisfy the one-prime power hypothesis have been classified‎. ‎However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups‎. ‎In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups‎.Sat, 30 Nov 2019 20:30:00 +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 +0100Further rigid triples of classes in $G_{2}$
http://ijgt.ui.ac.ir/article_23002_4261.html
We establish the existence of two rigid triples of conjugacy classes in the algebraic group G2 in characteristic 5, complementing results of the second author with Liebeck and Marion. As a corollary, the finite groups G2(5^n) are not (2,4,5)-generated, confirming a conjecture of Marion in this case.Sat, 30 Nov 2019 20:30:00 +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 +0100Graham Higman's PORC theorem
http://ijgt.ui.ac.ir/article_23003_4261.html
Graham Higman published two important papers in 1960‎. ‎In the first of these‎ ‎papers he proved that for any positive integer $n$ the number of groups of‎ ‎order $p^{n}$ is bounded by a polynomial in $p$‎, ‎and he formulated his famous‎ ‎PORC conjecture about the form of the function $f(p^{n})$ giving the number of‎ ‎groups of order $p^{n}$‎. ‎In the second of these two papers he proved that the‎ ‎function giving the number of $p$-class two groups of order $p^{n}$ is PORC‎. ‎He established this result as a corollary to a very general result about‎ ‎vector spaces acted on by the general linear group‎. ‎This theorem takes over a‎ ‎page to state‎, ‎and is so general that it is hard to see what is going on‎. ‎Higman's proof of this general theorem contains several new ideas and is quite‎ ‎hard to follow‎. ‎However in the last few years several authors have developed‎ ‎and implemented algorithms for computing Higman's PORC formulae in‎ ‎special cases of his general theorem‎. ‎These algorithms give perspective on‎ ‎what are the key points in Higman's proof‎, ‎and also simplify parts of the proof‎. ‎In this note I give a proof of Higman's general theorem written in the light‎ ‎of these recent developments‎.Sat, 30 Nov 2019 20:30:00 +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 +0100Limits of generalized quaternion groups
http://ijgt.ui.ac.ir/article_23004_4261.html
‎In the space of marked group‎s, ‎we determine the structure of groups which are limit points of the set of all generalized quaternion groups‎.Sat, 30 Nov 2019 20:30:00 +0100Classifying families of character degree graphs of solvable groups
http://ijgt.ui.ac.ir/article_23008_4261.html
‎We investigate prime character degree graphs of solvable groups‎. ‎In particular‎, ‎we consider a family of graphs $Gamma_{k,t}$ constructed by adjoining edges between two complete graphs in a one-to-one fashion‎. ‎In this paper we determine completely which graphs $Gamma_{k,t}$ occur as the prime character degree graph of a solvable group‎.Sat, 30 Nov 2019 20:30:00 +0100