International Journal of Group TheoryInternational Journal of Group Theory
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Fri, 16 Nov 2018 01:38:35 +0100FeedCreatorInternational Journal of Group Theory
http://ijgt.ui.ac.ir/
Feed provided by International Journal of Group Theory. Click to visit.Locally graded groups with a condition on infinite subsets
http://ijgt.ui.ac.ir/article_21234_4091.html
Let $G$ be a group‎, ‎we say that $G$ satisfies the property $mathcal{T}(infty)$ provided that‎, ‎every infinite set of elements of $G$ contains elements $xneq y‎, ‎z$ such that $[x‎, ‎y‎, ‎z]=1=[y‎, ‎z‎, ‎x]=[z‎, ‎x‎, ‎y]$‎. ‎We denote by $mathcal{C}$ the class of all polycyclic groups‎, ‎$mathcal{S}$ the class of all soluble groups‎, ‎$mathcal{R}$ the class of all residually finite groups‎, ‎$mathcal{L}$ the class of all locally graded groups‎, ‎$mathcal{N}_2$ the class of all nilpotent group of class at most two‎, ‎and $mathcal{F}$ the class of all finite groups‎. ‎In this paper‎, ‎first we shall prove that if $G$ is a finitely generated locally graded group‎, ‎then $G$ satisfies $mathcal{T}(infty)$ if and only if $G/Z_2(G)$ is finite‎, ‎and then we shall conclude that if $G$ is a finitely generated group in $mathcal{T}(infty)$‎, ‎then‎ ‎[Ginmathcal{L}Leftrightarrow Ginmathcal{R}Leftrightarrow Ginmathcal{S}Leftrightarrow Ginmathcal{C}Leftrightarrow Ginmathcal{N}_2mathcal{F}.]‎Fri, 30 Nov 2018 20:30:00 +0100On noninner automorphisms of finite $p$-groups that fix the center elementwise
http://ijgt.ui.ac.ir/article_22412_0.html
In this paper we show that every finite nonabelian $p$-group $G$ in which the Frattini subgroup $Phi(G)$ has order $leq p^5$ admits a noninner automorphism of order $p$ leaving the center $Z(G)$ elementwise fixed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least $p^8$.Tue, 13 Feb 2018 20:30:00 +0100Automorphisms of a finite $p$-group with cyclic Frattini subgroup
http://ijgt.ui.ac.ir/article_21219_4091.html
Let $G$ be a group and $Aut^{Phi}(G)$ denote the group of all automorphisms of $G$ centralizing $G/Phi(G)$ elementwise‎. ‎In this paper‎, ‎we characterize the finite $p$-groups $G$ with cyclic Frattini subgroup for which $|Aut^{Phi}(G):Inn(G)|=p$‎.Fri, 30 Nov 2018 20:30:00 +0100On the ranks of Fischer group $Fi_{24}^{\,\prime}$ and the Baby Monster group $\mathbb{B}$
http://ijgt.ui.ac.ir/article_22709_0.html
If $G$ is a finite group and $X$ a conjugacy class of‎ ‎elements of $G$‎, ‎then we define $rank(G{:}X)$ to be the minimum‎ ‎number of elements of $X$ generating $G$‎. ‎In the present article‎, ‎we‎ ‎determine the ranks for the Fischer's simple group $Fi_{24}^{prime}$‎ ‎and the baby monster group $mathbb{B}$‎.Fri, 29 Jun 2018 19:30:00 +0100Finite groups with the same conjugacy class sizes as a finite simple group
http://ijgt.ui.ac.ir/article_21236_0.html
For a finite group $H$‎, ‎let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$‎. ‎In this paper‎, ‎we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$‎, ‎then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$‎. ‎In particular‎, ‎we show that for some finite simple group $S$‎, ‎$G cong S times Z(G)$‎.Fri, 30 Nov 2018 20:30:00 +0100On embedding of partially commutative metabelian groups to matrix groups
http://ijgt.ui.ac.ir/article_21478_4091.html
‎The Magnus embedding of a free metabelian group induces the embedding of partially commutative metabelian group $S_Gamma$ in a group of matrices $M_Gamma$. Properties and the universal theory of the group $M_Gamma$ are studied.Fri, 30 Nov 2018 20:30:00 +0100Measuring cones and other thick subsets in free groups
http://ijgt.ui.ac.ir/article_21479_4091.html
In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they accept‎. ‎We show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoids‎. ‎These automata allow us to compute explicitly generating functions‎, ‎$lambda-$measures and Cesaro measure of thick monoids‎. ‎Also we improve the asymptotic classification of regular subsets in free groups‎.Fri, 30 Nov 2018 20:30:00 +0100Finite groups of the same type as Suzuki groups
http://ijgt.ui.ac.ir/article_21556_0.html
‎For a finite group $G$ and a positive integer $n$‎, ‎let $G(n)$ be the set of all elements in $G$ such that $x^{n}=1$‎. ‎The groups $G$ and $H$ are said to be of the same (order) type if $|G(n)|=|H(n)|$‎, ‎for all $n$‎. ‎The main aim of this paper is to show that if $G$ is a finite group of the same type as Suzuki groups $Sz(q)$‎, ‎where $q=2^{2m+1}geq 8$‎, ‎then $G$ is isomorphic to $Sz(q)$‎. ‎This addresses to the well-known J‎. ‎G‎. ‎Thompson's problem (1987) for simple groups‎.Thu, 13 Jul 2017 19:30:00 +0100The Maschke property for the Sylow $p$-sub\-groups of the symmetric group $S_{p^n}$
http://ijgt.ui.ac.ir/article_21610_4091.html
‎‎In this paper we prove that the Maschke property holds for coprime actions on some important classes of $p$-groups like‎: ‎metacyclic $p$-groups‎, ‎$p$-groups of $p$-rank two for $p>3$ and some weaker property holds in the case of regular $p$-groups‎. ‎The main focus will be the case of coprime actions on the iterated wreath product $P_n$ of cyclic groups of order $p$‎, ‎i.e‎. ‎on Sylow $p$-subgroups of the symmetric groups $S_{p^n}$‎, ‎where we also prove that a stronger form of the Maschke property holds‎. ‎These results contribute to a future possible classification of all $p$-groups with the Maschke property‎. ‎We apply these results to describe which normal partition subgroups of $P_n$ have a complement‎. ‎In the end we also describe abelian subgroups of $P_n$ of largest size‎.Fri, 30 Nov 2018 20:30:00 +0100Difference bases in dihedral groups
http://ijgt.ui.ac.ir/article_21612_0.html
A subset $B$ of a group $G$ is called a {em‎ ‎difference basis} of $G$ if each element $gin G$ can be written as the‎ ‎difference $g=ab^{-1}$ of some elements $a,bin B$‎. ‎The smallest‎ ‎cardinality $|B|$ of a difference basis $Bsubset G$ is called the {em‎ ‎difference size} of $G$ and is denoted by $Delta[G]$‎. ‎The fraction ‎‎‎$eth[G]:=Delta[G]/{sqrt{|G|}}$ is called the {em difference characteristic} of $G$‎. ‎We prove that for every $nin N$ the dihedral group‎ ‎$D_{2n}$ of order $2n$ has the difference characteristic‎ ‎$sqrt{2}leeth[D_{2n}]leqfrac{48}{sqrt{586}}approx1.983$‎. ‎Moreover‎, ‎if $nge 2cdot 10^{15}$‎, ‎then $eth[D_{2n}]<frac{4}{sqrt{6}}approx1.633$‎. ‎Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $le80$‎.Wed, 16 Aug 2017 19:30:00 +0100On algebraic geometry over completely simple semigroups
http://ijgt.ui.ac.ir/article_21975_0.html
We study equations over completely simple semigroups and describe the coordinate semigroups of irreducible algebraic sets for such semigroups.Fri, 20 Oct 2017 20:30:00 +0100Transitive $t$-designs and strongly regular graphs constructed from linear groups $L(2,q)$, ...
http://ijgt.ui.ac.ir/article_21613_0.html
‎In this paper we construct transitive $t$-designs from the linear groups $L(2,q)‎, ‎q leq 23$‎. ‎Thereby we classify $t$-designs‎, ‎$t ge 2$‎, ‎admitting a transitive action of the linear groups $L(2,q)‎, ‎q leq 23$‎, ‎up to 35 points and obtained numerous transitive designs‎, ‎for $36leq vleq 55$‎. ‎In many cases we proved the existence of $t$-designs with certain parameter sets‎. ‎Among others we constructed $t$-designs with parameters $2$-$(55,10,4)$‎, ‎$3$-$(24,11,495)$‎, ‎$3$-$(24,12‎, ‎5m)‎, ‎m in {11‎, ‎12,22‎, ‎33‎, ‎44‎, ‎66‎, ‎132}$‎. ‎Furthermore‎, ‎we constructed strongly regular graphs admitting a transitive action of the linear groups $L(2,q)‎, ‎q leq 23$‎.Wed, 16 Aug 2017 19:30:00 +0100A classification of nilpotent $3$-BCI groups
http://ijgt.ui.ac.ir/article_22202_0.html
‎Given a finite group $G$ and a subset $Ssubseteq G,$ the bi-Cayley graph $BCay(G,S)$ is the graph whose vertex‎ ‎set is $G times {0,1}$ and edge set is‎ ‎${ {(x,0),(s x,1)}‎ : ‎x in G‎, ‎sin S }$‎. ‎A bi-Cayley graph $BCay(G,S)$ is called a BCI-graph if for any bi-Cayley graph‎ ‎$BCay(G,T),$ $BCay(G,S) cong BCay(G,T)$ implies that $T = g S^alpha$ for some $g in G$ and $alpha in Aut(G)$‎. ‎A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs‎. ‎It was proved by Jin and Liu that‎, ‎if $G$ is a $3$-BCI-group‎, ‎then its Sylow $2$-subgroup is cyclic‎, ‎or elementary abelian‎, ‎or $Q$ [European J‎. ‎Combin‎. ‎31 (2010)‎ ‎1257--1264]‎, ‎and that a Sylow $p$-subgroup‎, ‎$p$ is an odd prime‎, ‎is homocyclic [Util‎. ‎Math‎. ‎86 (2011) 313--320]‎. ‎In this paper we show that the converse also holds in the‎ ‎case when $G$ is nilpotent‎, ‎and hence complete the classification of‎ ‎nilpotent $3$-BCI-groups‎.Mon, 11 Dec 2017 20:30:00 +0100${\rm B}_\pi$-characters and quotients
http://ijgt.ui.ac.ir/article_22203_0.html
Let $pi$ be a set of primes, and let $G$ be a finite $pi$-separable group. We consider the Isaacs ${rm B}_pi$-characters. We show that if $N$ is a normal subgroup of $G$, then ${rm B}_pi (G/N) = irr {G/N} cap {rm B}_pi (G)$.Mon, 11 Dec 2017 20:30:00 +0100On free subgroups of finite exponent in circle groups of free nilpotent algebras
http://ijgt.ui.ac.ir/article_22208_0.html
Let $K$ be a commutative ring with identity and $N$ the free nilpotent $K$-algebra on a non-empty set $X$. Then $N$ is a group with respect to the circle composition. We prove that the subgroup generated by $X$ is relatively free in a suitable class of groups, depending on the choice of $K$. Moreover, we get unique representations of the elements in terms of basic commutators. In particular, if $K$ is of characteristic $0$ the subgroup generated by $X$ is freely generated by $X$ as a nilpotent group.Fri, 15 Dec 2017 20:30:00 +0100Recognition of the simple groups $PSL_2(q)$ by character degree graph and order
http://ijgt.ui.ac.ir/article_22212_0.html
‎Let $G$ be a finite group‎, ‎and $Irr(G)$ be the set of complex irreducible characters of $G$‎. ‎Let $rho(G)$ be the set of prime divisors of character degrees of $G$‎. ‎The character degree graph of $G$‎, ‎which is denoted by $Delta(G)$‎, ‎is a simple graph with vertex set $rho(G)$‎, ‎and we join two vertices $r$ and $s$ by an edge if there exists a character degree of $G$ divisible by $rs$‎. ‎In this paper‎, ‎we prove that if $G$ is a finite group such that $Delta(G)=Delta(PSL_2(q))$ and $|G|=|PSL_2(q)|$‎, ‎then $G cong PSL_2(q)$‎.Sat, 16 Dec 2017 20:30:00 +0100On finite groups having a certain number of cyclic subgroups
http://ijgt.ui.ac.ir/article_22472_0.html
Let $G$ be a finite group. In this paper, we study the structure of finite groups having $|G|-r$ cyclic subgroups for $3leq rleq 5$.Sun, 11 Mar 2018 20:30:00 +0100A note on Engel elements in the first Grigorchuk group
http://ijgt.ui.ac.ir/article_22725_0.html
Let $Gamma$ be the first Grigorchuk group‎. ‎According to a result of Bar-thol-di‎, ‎the only left Engel elements of $Gamma$ are the involutions‎. ‎This implies that the set of left Engel elements of $Gamma$ is not a subgroup‎. ‎The natural question arises whether this is also the case for the sets of bounded left Engel elements‎, ‎right Engel elements and bounded right Engel elements of $Gamma$‎. ‎Motivated by this‎, ‎we prove that these three subsets of $Gamma$ coincide with the identity subgroup‎.Tue, 03 Jul 2018 19:30:00 +0100Upper bounds on the uniform spreads of the sporadic simple groups
http://ijgt.ui.ac.ir/article_22875_0.html
‎A finite group $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ of elements in $G$ with the property that‎ ‎for any $k$ nontrivial elements $s_1, s_2,‎ldots‎,s_k$ in $G$ there exists $yin C$ such that $G = langle s_i,yrangle$ for $i=1, 2,‎ldots,k$‎. ‎Further‎, ‎the exact uniform spread of $G$ is the largest $k$ such that $G$ has the uniform spread $k$‎. ‎In this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups‎.Mon, 03 Sep 2018 19:30:00 +0100A presentation for the subgroup of compressed conjugating automorphisms of a partially ...
http://ijgt.ui.ac.ir/article_23000_0.html
Let G be a partially commutative group. We construct a finite presentation for the subgroup Conjv(G ) of compressed vertex conjugating automorphisms of the automorphism group Aut(G) of G. We have written GAP packages which compute presentations for Aut(G) and for its subgroups of conjugating automorphisms Conj(G) and for Conjv(G).Mon, 15 Oct 2018 20:30:00 +0100The one-prime power hypothesis for conjugacy classes restricted to normal subgroups and ...
http://ijgt.ui.ac.ir/article_23001_0.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.Mon, 15 Oct 2018 20:30:00 +0100Further Rigid Triples of Classes in G_2
http://ijgt.ui.ac.ir/article_23002_0.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.Mon, 15 Oct 2018 20:30:00 +0100Graham Higman's PORC theorem
http://ijgt.ui.ac.ir/article_23003_0.html
‎Graham Higman published two important papers in 1960‎. ‎In the first of these‎‎papers he proved that for any positive integer $n$ the number of groups of‎‎order $p^{n}$ is bounded by a polynomial in $p$‎, ‎and he formulated his famous‎‎PORC conjecture about the form of the function $f(p^{n})$ giving the number of‎‎groups of order $p^{n}$‎. ‎In the second of these two papers he proved that the‎‎function 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 about‎‎vector spaces acted on by the general linear group‎. ‎This theorem takes over a‎‎page 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 quite‎‎hard to follow‎. ‎However in the last few years several authors have developed‎‎and implemented algorithms for computing Higman's PORC formulae in‎‎special cases of his general theorem‎. ‎These algorithms give perspective on‎‎what 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 light‎‎of these recent developments‎.Mon, 15 Oct 2018 20:30:00 +0100Limits of generalized quaternion groups
http://ijgt.ui.ac.ir/article_23004_0.html
In the space of marked group, we determine the structure of groups which are limit points of the set of all generalized quaternion groups.Mon, 15 Oct 2018 20:30:00 +0100Classifying families of character degree graphs of solvable groups
http://ijgt.ui.ac.ir/article_23008_0.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.Tue, 16 Oct 2018 20:30:00 +0100