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 groups with square-free conjugacy class sizes
http://ijgt.ui.ac.ir/article_21475_0.html
We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.Wed, 01 Mar 2017 20:30:00 +0100The conjugacy class ranks of $M_{24}$
http://ijgt.ui.ac.ir/article_21477_0.html
$M_{24}$ is the largest Mathieu sporadic simple group of order $244 823 040 = 2^{10} {cdot} 3^3 {cdot} 5 {cdot} 7 {cdot} 11 {cdot} 23$ and contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of $M_{24}$ with respect to the conjugacy classes of all its nonidentity elements.Sun, 14 May 2017 19:30:00 +0100On embedding of partially commutative metabelian groups to matrix groups
http://ijgt.ui.ac.ir/article_21478_0.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.Thu, 08 Jun 2017 19:30:00 +0100On nonsolvable groups whose prime degree graphs have four vertices and one triangle
http://ijgt.ui.ac.ir/article_21476_0.html
‎Let $G$ be a finite group‎. ‎The prime degree graph of $G$‎, ‎denoted‎ ‎by $Delta(G)$‎, ‎is an undirected graph whose vertex set is $rho(G)$ and there is an edge‎ ‎between two distinct primes $p$ and $q$ if and only if $pq$ divides some irreducible‎ ‎character degree of $G$‎. ‎In general‎, ‎it seems that the prime graphs‎ ‎contain many edges and thus they should have many triangles‎, ‎so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles‎. ‎In this paper we consider the case where for a nonsolvable group $G$‎, ‎$Delta(G)$ is a connected graph which has only one triangle and four vertices‎.Thu, 08 Jun 2017 19:30:00 +0100Measuring cones and other thick subsets in free groups
http://ijgt.ui.ac.ir/article_21479_0.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.Thu, 08 Jun 2017 19:30:00 +0100On metacyclic subgroups of finite groups
http://ijgt.ui.ac.ir/article_21480_0.html
The aim of this survey article is to present some structural results about of groups whose Sylow $p$-subgroups are metacylic ($p$ a prime). A complete characterisation of non-nilpotent groups whose $2$-generator subgroups are metacyclic is also presented.Thu, 08 Jun 2017 19:30:00 +0100On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras
http://ijgt.ui.ac.ir/article_21481_0.html
Let $L$ be a free Lie algebra of rank $rgeq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$‎. ‎By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$‎, ‎we determine the dimension of $[L_2,L_2,L_1]$‎. ‎Moreover‎, ‎by this method‎, ‎we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.Thu, 08 Jun 2017 19:30:00 +0100Sylow multiplicities in finite groups
http://ijgt.ui.ac.ir/article_21482_0.html
Let $G$ be a finite group and let $mathcal{P}=P_{1},ldots,P_{m}$ be a sequence‎ ‎of Sylow $p_{i}$-subgroups of $G$‎, ‎where $p_{1},ldots,p_{m}$ are the distinct‎ ‎prime divisors of $leftvert Grightvert $‎. ‎The Sylow multiplicity of $gin‎ ‎G$ in $mathcal{P}$ is the number of distinct factorizations $g=g_{1}cdots‎ ‎g_{m}$ such that $g_{i}in P_{i}$‎. ‎We review properties of the solvable‎ ‎radical and the solvable residual of $G$ which are formulated in terms of‎ ‎Sylow multiplicities‎, ‎and discuss some related open questions‎.Thu, 08 Jun 2017 19:30:00 +0100Groups with permutability conditions for subgroups of infinite rank
http://ijgt.ui.ac.ir/article_21483_0.html
In this paper, the structure of non-periodic generalized radical groups of infinite rank whose subgroups of infinite rank satisfy a suitable permutability condition is investigated.Thu, 08 Jun 2017 19:30:00 +0100Representations of group rings and groups
http://ijgt.ui.ac.ir/article_21484_0.html
An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix $A$ of $mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,ldots, T_r)$ for block matrices $T_i$ of fixed size $s_iti s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ are the ranks of the group ring matrices of the primitive idempotents. Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping $Amapsto P^{-1}AP$ (fixed $P$) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of $G$ may be derived using the blocks in the images of the group elements. For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $mathbb{C}Q$. The characters of $Q$ and the character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing.Thu, 08 Jun 2017 19:30:00 +0100An infinite family of finite $2$-groups with deficiency zero
http://ijgt.ui.ac.ir/article_21213_0.html
‎‎We determine a new infinite sequence of finite $2$-groups with deficiency zero‎. ‎The groups have $2$ generators and $2$ relations‎, ‎they have coclass $3$‎ ‎and they are not metacyclic‎.Fri, 03 Mar 2017 20:30:00 +0100Some characterisations of groups in which normality is a transitive relation by means of ...
http://ijgt.ui.ac.ir/article_21214_0.html
‎In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation in‎ ‎certain universes of groups with some finiteness properties‎.Fri, 03 Mar 2017 20:30:00 +0100Regular subgroups, nilpotent algebras and projectively congruent matrices
http://ijgt.ui.ac.ir/article_21215_0.html
‎In this paper we highlight the connection between certain classes of regular subgroups of the affine group‎ ‎$AGL_n(F)$‎, ‎$F$ a field‎, ‎and associative nilpotent $F$-algebras of dimension $n$‎. ‎We also describe how the classification of projective congruence classes of square matrices is equivalent to the‎ ‎classification of regular subgroups of particular shape‎.Tue, 14 Feb 2017 20:30:00 +0100Conjugacy classes contained in normal subgroups: an overview
http://ijgt.ui.ac.ir/article_21216_0.html
We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an influence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau's Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions.Mon, 16 Jan 2017 20:30:00 +0100On the relationships between the factors of the upper and lower central series in some ...
http://ijgt.ui.ac.ir/article_21217_0.html
This paper deals with the mutual relationships between the factor group $G/zeta(G)$ (respectively $G/zeta_k(G)$) and $G'$ (respectively $gamma_{k+1}(G)$ and $G^{mathfrak{N}}$)‎. ‎It is proved that if $G/zeta(G)$ (respectively $G/zeta_k(G)$) has finite $0$-rank‎, ‎then $G'$ (respectively $gamma_{k+1}(G)$ and $G^{mathfrak{N}}$) also have finite $0$-rank‎. ‎Furthermore‎, ‎bounds for the $0$-ranks of $G'‎, ‎gamma_{k+1}(G)$ and $G^{mathfrak{N}}$ are obtained‎.Mon, 19 Dec 2016 20:30:00 +0100On groups with two isomorphism classes of central factors
http://ijgt.ui.ac.ir/article_21218_0.html
The structure of groups which have at most two isomorphism classes of central factors ($B_2$-groups) are investigated‎. ‎A complete description of $B_2$-groups is obtained in the locally finite case and in the nilpotent case‎. ‎In addition detailed information is obtained about soluble $B_2$-groups‎. ‎Also structural information about insoluble $B_2$-groups is given‎, ‎in particular when such a group has the derived subgroup satisfying the minimal condition‎.Wed, 30 Nov 2016 20:30:00 +0100Finite $2$-groups of class $2$ with a specific automorphism group
http://ijgt.ui.ac.ir/article_20362_3873.html
‎‎In this paper we determine all finite $2$-groups of‎ ‎class $2$ in which every automorphism of order $2$ leaving the Frattini subgroup elementwise fixed is inner‎.Thu, 31 Aug 2017 19:30:00 +0100Automorphisms of a finite $p$-group with cyclic Frattini subgroup
http://ijgt.ui.ac.ir/article_21219_0.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, 06 Jan 2017 20:30:00 +0100Detecting the prime divisors of the character degrees and the class sizes by a subgroup ...
http://ijgt.ui.ac.ir/article_21220_0.html
We prove that every finite group $G$ contains a three-generated subgroup $H$ with the following property‎: ‎a prime $p$ divides the degree of an irreducible character of $G$ if and only if it divides the degree of an irreducible character of $H.$ There is no analogous result for the prime divisors of the sizes of the conjugacy classes‎.Sun, 12 Feb 2017 20:30:00 +0100An extension and a generalization of Dedekind's theorem
http://ijgt.ui.ac.ir/article_21238_3873.html
For any given finite abelian group‎, ‎we give factorizations of the group determinant in the group algebra of any subgroups‎. ‎The factorizations is an extension of Dedekind's theorem‎. ‎The extension leads to a generalization of Dedekind's theorem‎.Thu, 31 Aug 2017 19:30:00 +0100Bipartite divisor graph for the set of irreducible character degrees
http://ijgt.ui.ac.ir/article_21221_0.html
‎Let $G$ be a finite group‎. ‎We consider the set of the irreducible complex characters of $G$‎, ‎namely $Irr(G)$‎, ‎and the related degree set $cd(G)={chi(1)‎ : ‎chiin Irr(G)}$‎. ‎Let $rho(G)$ be the set of all primes which divide some character degree of $G$‎. ‎In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $rho(G)cup (cd(G)setminus{1})$‎, ‎such that an element $p$ of $rho(G)$ is adjacent to an element $m$ of $cd(G)setminus{1}$ if and only if $p$ divides $m$‎. ‎We denote this graph simply by $B(G)$‎. ‎Then by means of combinatorial properties of this graph‎, ‎we discuss the structure of the group $G$‎. ‎In particular‎, ‎we consider the cases where $B(G)$ is a path or a cycle‎.Fri, 03 Feb 2017 20:30:00 +0100Finite non-nilpotent groups with few non-normal non-cyclic subgroups
http://ijgt.ui.ac.ir/article_21222_0.html
‎‎For a finite group $G$‎, ‎let $nu_{nc}(G)$ denote the number of conjugacy classes of non-normal non-cyclic subgroups of $G$‎. ‎We characterize the finite non-nilpotent groups whose all non-normal non-cyclic subgroups are conjugate‎.Wed, 11 Jan 2017 20:30:00 +0100One-prime power hypothesis for conjugacy class sizes
http://ijgt.ui.ac.ir/article_12043_0.html
A finite group $G$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to $S times A$ where $A$ is abelian and $S cong PSL_2(q)$ for $q in {4,8}$. We confirm this conjecture.Thu, 17 Dec 2015 20:30:00 +0100On almost recognizability by spectrum of simple classical groups
http://ijgt.ui.ac.ir/article_21223_0.html
‎The set of element orders of a finite group $G$ is called the {em spectrum}‎. ‎Groups with coinciding spectra are said to be {em isospectral}‎. ‎It is known that if $G$ has a nontrivial normal soluble subgroup then there exist infinitely many pairwise non-isomorphic‎ ‎groups isospectral to $G$‎. ‎The situation is quite different if $G$ is a nonabelain simple group‎. ‎Recently it was proved that if $L$ is a simple classical group of dimension at least 62 and $G$ is a finite group‎ ‎isospectral to $L$‎, ‎then up to isomorphism $Lleq GleqAut L$‎. ‎We show that the assertion remains true‎ ‎if 62 is replaced by 38‎.Sat, 10 Dec 2016 20:30:00 +0100A new characterization of Ree group $\mathbf{{}^2G_2(q)}$ by the order of group and the number ...
http://ijgt.ui.ac.ir/article_21233_0.html
‎‎In this paper‎, ‎we prove that Ree group ${}^2G_2(q)$‎, ‎where $qpmsqrt{3q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order‎.Thu, 24 Nov 2016 20:30:00 +0100Locally Graded groups with a condition on infinite subsets
http://ijgt.ui.ac.ir/article_21234_0.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}.]Sat, 18 Jun 2016 19:30:00 +0100Countably recognizable classes of groups with restricted conjugacy classes
http://ijgt.ui.ac.ir/article_21235_0.html
A group class {mgoth X} is said to be countably recognizable if a group belongs to {mgoth X} whenever all its countable subgroups lie in {mgoth X}‎. ‎It is proved here that most of the relevant classes of groups defined by restrictions on the conjugacy classes are countably recognizable‎.Fri, 26 Aug 2016 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, 03 Mar 2017 20:30:00 +0100On groups with a restriction on normal subgroups
http://ijgt.ui.ac.ir/article_21237_0.html
The structure of infinite groups in which every (proper) normal subgroup is the only one of its cardinality is investigated in the universe of groups without infinite simple sections‎. ‎The corrisponding problem for finite soluble groups was considered by M‎. ‎Curzio (1958)‎.Tue, 23 Aug 2016 19:30:00 +0100Right amenable left group sets and the Tarski-FØlner theorem
http://ijgt.ui.ac.ir/article_21243_0.html
‎We introduce right amenability‎, ‎right FØlner nets‎, ‎and right paradoxical decompositions for left homogeneous spaces and prove the Tarski-FØlner theorem for left homogeneous spaces with finite stabilisers‎. ‎It states that right amenability‎, ‎the existence of right FØlner nets‎, ‎and the non-existence of right paradoxical decompositions are equivalent‎.Thu, 02 Mar 2017 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 the well-known J‎. ‎G‎. ‎Thompson's problem (1987) for simple groups‎.Thu, 13 Jul 2017 19:30:00 +0100Finite groups with non-trivial intersections of kernels of all but one irreducible characters
http://ijgt.ui.ac.ir/article_21609_0.html
In this paper we consider finite groups $G$ satisfying the following‎ ‎condition‎: ‎$G$ has two columns in its character table which differ by exactly one‎ ‎entry‎. ‎It turns out that such groups exist and they are exactly the finite groups‎ ‎with a non-trivial intersection of the kernels of all but one irreducible‎ ‎characters or‎, ‎equivalently‎, ‎finite groups with an irreducible character‎ ‎vanishing on all but two conjugacy classes‎. ‎We investigate such groups‎ ‎and in particular we characterize their subclass‎, ‎which properly contains‎ ‎all finite groups with non-linear characters of distinct degrees‎, ‎which were characterized by Berkovich‎, ‎Chillag and Herzog in 1992‎.Wed, 16 Aug 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_0.html
‎In this paper we prove that 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 iterated wreath product $P_n$‎ ‎of cyclic groups of order $p$‎, ‎i.e‎. ‎on Sylow $p$-subgroups of symmetric groups $S_{p^n}$‎, ‎where we also prove that a stronger form of Maschke property holds‎. ‎These results contribute to a future possible classification for all $p$-groups with Maschke property‎. ‎We apply these results to describe which normal partition subgroups of $P_n$‎ ‎have complement‎. ‎In the end we also describe abelian subgroups of largest size in $P_n$‎.Wed, 16 Aug 2017 19:30:00 +0100Inertial properties in groups
http://ijgt.ui.ac.ir/article_21611_0.html
‎Let $G$ be a group and $p$ be an endomorphism of $G$‎. ‎A subgroup $H$ of $G$ is called {em $p$-inert} if $H^pcap H$ has finite index in the image $H^p$‎. ‎The subgroups that are {em $p$-inert} for all inner automorphisms of $G$ are widely known and studied in the literature‎, ‎under the name {em inert} subgroups‎.
‎The related notion of {em inertial endomorphism}‎, ‎namely an endomorphism $p$ such that all subgroups of $G$ are {$p$-inert}‎, ‎was introduced in [33] and thoroughly studied in [34,36]. ‎The ``dual‎" ‎notion of {em fully inert subgroup}‎, ‎namely a subgroup that is {em $p$-inert} for all endomorphisms of an abelian group $A$‎, ‎was introduced in [50] and further studied in [24,53,75]‎.
‎
‎The goal of this paper is to give an overview of up-to-date known results‎, ‎as well as some new ones‎, ‎and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra‎. ‎We survey on classical and recent results on groups whose inner automorphisms are inertial‎. ‎Moreover‎, ‎we show how ‎‎‎‎inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces‎, ‎and can be helpful for the computation of the algebraic entropy of continuous endomorphisms‎.Wed, 16 Aug 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 $ninIN$ 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 +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 +0100