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_4453.html
Let $gamma_n=[x_1,ldots,x_n]$ be the $n$th lower central word‎. ‎Suppose that $G$ is a profinite group‎ ‎where the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$‎ ‎elements‎ ‎for 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 that‎ ‎a 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, 30 Nov 2020 20:30:00 +0100Algorithmic problems in Engel groups and cryptographic applications
http://ijgt.ui.ac.ir/article_24453_4453.html
‎The theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside problems‎. ‎In this survey we consider several classical and novel algorithmic problems for Engel groups and propose several open problems‎. ‎We study these problems with a view towards applications to cryptography‎.Mon, 30 Nov 2020 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 +0100Engel groups in bath - ten years later
http://ijgt.ui.ac.ir/article_24420_4453.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.Mon, 30 Nov 2020 20:30:00 +0100Groups with many roots
http://ijgt.ui.ac.ir/article_24499_4453.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 $myro_3$‎.Mon, 30 Nov 2020 20:30:00 +0100The fibonacci-circulant sequences in the binary polyhedral groups
http://ijgt.ui.ac.ir/article_24424_0.html
In 2017 Deveci et al‎. ‎defined 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 paper‎, ‎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 +0100Small doubling in $m$-Engel groups
http://ijgt.ui.ac.ir/article_24532_4453.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 $b$‎.Mon, 30 Nov 2020 20:30:00 +0100Some remarks on unipotent automorphisms
http://ijgt.ui.ac.ir/article_24519_4453.html
An automorphism $alpha$ of the group $G$ is said to be $n$-unipotent if $[g,_nalpha]=1$ for all $gin G$‎. ‎In this paper we obtain some results related to nilpotency of groups of $n$-unipotent automorphisms of solvable groups‎. ‎We also show that‎, ‎assuming the truth of a conjecture about the representation theory of solvable groups raised by P‎. ‎Neumann‎, ‎it is possible to produce‎, ‎for a suitable prime $p$‎, ‎an example of a f.g‎. ‎solvable group possessing a group of $p$-unipotent automorphisms which is isomorphic to an infinite Burnside group‎. ‎Conversely we show that‎, ‎if there exists a f.g‎. ‎solvable group $G$ with a non nilpotent $p$-group $H$ of $n$-automorphisms‎, ‎then there is such a counterexample where $n$ is a prime power and $H$ has finite exponent‎.Mon, 30 Nov 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 +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 ‎investigate the power graphs of three‎ ‎classes of finite groups‎: ‎the elementary‎ ‎abelian groups of exponent $p$‎, ‎and the‎ ‎extra special groups of exponents $p$ or‎ ‎$p^2$‎. ‎We show that these power graphs‎ ‎are Eulerian for every $p$‎. ‎As a‎ ‎corollary‎, ‎we describe two classes of‎ ‎non-isomorphic groups with isomorphic‎ ‎power graphs‎. ‎In addition‎, ‎we prove that‎ ‎the clique graphs of the power graphs of‎ ‎two considered classes are complete‎.Tue, 24 Mar 2020 19:30:00 +0100The probability of commuting subgroups in arbitrary lattices of subgroups
http://ijgt.ui.ac.ir/article_24551_0.html
A finite group $G$‎, ‎in which two randomly chosen subgroups $H$ and $K$ commute‎, ‎has been classified by Iwasawa in 1941‎. ‎It is possible to define a probabilistic notion‎, ‎which ``measures the distance'' of $G$ from the groups of Iwasawa‎. ‎Here we introduce the generalized subgroup commutativity degree $gsd(G)$ for two arbitrary sublattices $mathrm{S}(G)$ and $mathrm{T}(G)$ of the lattice of subgroups $mathrm{L}(G)$ of $G$‎. ‎Upper and lower bounds for $gsd(G)$ are shown and we study the behaviour of $gsd(G)$ with respect to subgroups and quotients‎, ‎showing new numerical restrictions‎.Thu, 16 Apr 2020 19:30:00 +0100Maximal abelian subgroups of the finite symmetric group
http://ijgt.ui.ac.ir/article_24559_0.html
‎Let $G$ be a group‎. ‎For an element $ain G$‎, ‎denote by $cs(a)$ the‎ ‎second centralizer of~$a$ in~$G$‎, ‎which is the set of all elements $bin G$‎ ‎such that $bx=xb$ for every $xin G$ that commutes with $a$‎. ‎Let $M$ be any maximal abelian subgroup of $G$‎. ‎Then $cs(a)subseteq M$ for every $ain M$‎. ‎The emph{abelian rank} (emph{$a$-rank}) of $M$‎ ‎is the minimum cardinality of a set $Asubseteq M$ such that $bigcup_{ain A}cs(a)$ generates $M$‎. ‎Denote by $S_n$ the symmetric group of permutations on the set $X={1,ldots,n}$‎. ‎The aim of this paper is to determine the maximal abelian subgroups of $gx$‎‎of $cor$~$1$ and describe a class of maximal abelian‎ ‎subgroups of $gx$ of $cor$ at most~$2$‎.Fri, 24 Apr 2020 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 +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 +0100Open problems from the conference "Engel Conditions in Groups" held in ...
http://ijgt.ui.ac.ir/article_24830_4453.html
Here is list of open problems from the conference Engel Type Conditions in Groups in Bath that was held in April 2019.Mon, 30 Nov 2020 20:30:00 +0100Parameters of the Coprime Graph of a Group
http://ijgt.ui.ac.ir/article_24696_0.html
‎There are many different graphs one can associate to a group‎. ‎Some examples are the well-known Cayley graph‎, ‎the zero divisor graph (of a ring)‎, ‎the power graph‎, ‎and the recently introduced coprime graph of a group‎. ‎The coprime graph of a group $G$‎, ‎denoted $Gamma_G$‎, ‎is the graph whose vertices are the group elements with $g$ adjacent to $h$ if and only if $(o(g),o(h))=1$‎. ‎In this paper we calculate the independence number of the coprime graph of the dihedral groups‎. ‎Additionally‎, ‎we characterize the groups whose coprime graph is perfect‎.Mon, 15 Jun 2020 19:30:00 +0100The automorphism groups of groups of order $p^{2} q$
http://ijgt.ui.ac.ir/article_24841_0.html
‎We record for reference a detailed description of the automorphism‎ ‎groups of the groups of order $p^{2}q$‎, ‎where $p$ and $q$ are‎ ‎distinct primes‎.Tue, 28 Jul 2020 19:30:00 +0100Subgroups of arbitrary even ordinary depth
http://ijgt.ui.ac.ir/article_24854_0.html
‎We show that for each positive integer $n$‎, ‎there exist a group $G$ and a subgroup $H$ such that the ordinary depth‎ ‎$d(H‎, ‎G)$ is $2n$‎. ‎This solves the open problem posed by Lars Kadison ‎‎‎‎whether even ordinary depth larger than $6$ can occur‎.Tue, 04 Aug 2020 19:30:00 +0100Schur's exponent conjecture - counterexamples of exponent 5 and exponent 9
http://ijgt.ui.ac.ir/article_24899_0.html
There is a long-standing conjecture attributed to I. Schur that if G is a finite group with Schur multiplier M(G) then the exponent of M(G) divides the exponent of G. In this note I give an example of a four generator group G of order 5^{4122} with exponent 5, where the Schur multiplier M(G) has exponent 25.Tue, 18 Aug 2020 19:30:00 +0100Some results on the join graph of finite groups
http://ijgt.ui.ac.ir/article_25026_0.html
Let $G$ be a finite group which is not cyclic of prime power order.The join graph $Delta(G)$ of $G$ isa graph whose vertex set is the set of all proper subgroups of $G$, which are not containedin the Frattini subgroup $G$ and two distinct vertices$H$ and $K$ are adjacent if and only if $G=langle H, Krangle$.Among other results, we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $Delta(G)congDelta(H)$, then $H$ is cyclic.Also we prove that $Delta(G)congDelta(A_5)$ if and only if $Gcong A_5$.Sun, 04 Oct 2020 20:30:00 +0100Boundedly finite conjugacy classes of tensors
http://ijgt.ui.ac.ir/article_25060_0.html
Let $n$ be a positive integer and let $G$ be a group. We denote by $nu(G)$ a certain extension of the non-abelian tensor square $G otimes G$ by $G times G$. Set $T_{otimes}(G) = {g otimes h mid g,h in G}$. We prove that if the size of the conjugacy class $left |x^{nu(G)} right| leq n$ for every $x in T_{otimes}(G)$, then the second derived subgroup $nu(G)''$ is finite with $n$-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.Mon, 19 Oct 2020 20:30:00 +0100