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.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 +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 +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 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 +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 +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 +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 +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 +0100