University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
6
1
2017
03
01
On bipartite divisor graph for character degrees
1
7
EN
Seyed Ali
Moosavi
University of Qom
s.a.mousavi@qom.ac.ir
The concept of the bipartite divisor graph for integer subsets has been considered in [M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, {em Graphs Combin.}, {bf 26} (2010) 95--105.]. In this paper, we will consider this graph for the set of character degrees of a finite group $G$ and obtain some properties of this graph. We show that if $G$ is a solvable group, then the number of connected components of this graph is at most $2$ and if $G$ is a non-solvable group, then it has at most $3$ connected components. We also show that the diameter of a connected bipartite divisor graph is bounded by $7$ and obtain some properties of groups whose graphs are complete bipartite graphs.
bipartite divisor graph,character degree,connected component,diameter
http://ijgt.ui.ac.ir/article_9852.html
http://ijgt.ui.ac.ir/article_9852_9330be05a79da3999795a5098e2d1f78.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
6
1
2017
03
01
Lipschitz groups and Lipschitz maps
9
16
EN
Laurent
Poinsot
University Paris 13, Paris Sorbonne Cité
laurent.poinsot@lipn.univ-paris13.fr
This contribution mainly focuses on some aspects of Lipschitz groups, i.e., metrizable groups with Lipschitz multiplication and inversion map. In the main result it is proved that metric groups, with a translation-invariant metric, may be characterized as particular group objects in the category of metric spaces and Lipschitz maps. Moreover, up to an adjustment of the metric, any metrizable abelian group also is shown to be a Lipschitz group. Finally we present a result similar to the fact that any topological nilpotent element $x$ in a Banach algebra gives rise to an invertible element $1-x$, in the setting of complete Lipschitz groups.
Lipschitz maps,group object in a category,translation-invariant metric
http://ijgt.ui.ac.ir/article_10506.html
http://ijgt.ui.ac.ir/article_10506_0454ef2c3bfb84c29f58794c6552fea9.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
6
1
2017
03
01
Shen's conjecture on groups with given same order type
17
20
EN
Leyli
Jafari Taghvasani
Department of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, Iran
l.jafari@sci.uok.ac.ir
Mohammad
Zarrin
University of Kurdistan
zarrin@ipm.ir
For any group $G$, we define an equivalence relation $thicksim$ as below: [forall g, h in G gthicksim h Longleftrightarrow |g|=|h|] the set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$ and denote by $alpha{(G)}$. In this paper, we give a partial answer to a conjecture raised by Shen. In fact, we show that if $G$ is a nilpotent group, then $|pi(G)|leq |alpha{(G)}|$, where $pi(G)$ is the set of prime divisors of order of $G$. Also we investigate the groups all of whose proper subgroups, say $H$ have $|alpha{(H)}|leq 2$.
Nilpotent groups,Same-order type,Schmidt group
http://ijgt.ui.ac.ir/article_10631.html
http://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
6
1
2017
03
01
A characterization of soluble groups in which normality is a transitive relation
21
27
EN
Giovanni
Vincenzi
University of Salerno
vincenzi@unisa.it
A subgroup $X$ of a group $G$ is said to be an {it{scriptsizecalligra H} -subgroup} if $N_G(X)cap X^gleq X$ for each element $g$ belonging to $G$. In [M. Bianchi and e.a., On finite soluble groups in which normality is a transitive relation, {em J. Group Theory}, {bf 3} (2000) 147--156.] the authors showed that finite groups in which every subgroup has the {scriptsizecalligra H} -property are exactly soluble groups in which normality is a transitive relation. Here we extend this characterization to groups without simple sections.
{{scriptsizecalligra H} -subgroups, {scriptsizecalligra T} -groups, pronormal subgroups,weakly normal subgroups,pronorm and {scriptsizecalligra H} -norm of a group.}
http://ijgt.ui.ac.ir/article_10890.html
http://ijgt.ui.ac.ir/article_10890_1b9c272898954a2b55df94ab7233e6e9.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
6
1
2017
03
01
Groups for which the noncommuting graph is a split graph
29
35
EN
Marzieh
Akbari
K. N. Toosi University of Technology
m.akbari@dena.kntu.ac.ir
Alireza
Moghaddamfar
K.N. Toosi University of Technology
moghadam@kntu.ac.ir
The noncommuting graph $nabla (G)$ of a group $G$ is a simple graph whose vertex set is the set of noncentral elements of $G$ and the edges of which are the ones connecting two noncommuting elements. We determine here, up to isomorphism, the structure of any finite nonabeilan group $G$ whose noncommuting graph is a split graph, that is, a graph whose vertex set can be partitioned into two sets such that the induced subgraph on one of them is a complete graph and the induced subgraph on the other is an independent set.
nonabelian group,noncommuting graph,split graph
http://ijgt.ui.ac.ir/article_11161.html
http://ijgt.ui.ac.ir/article_11161_c917794bead2c36c83dfeb54b217fb49.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
6
1
2017
03
01
Torsion units for some projected special linear groups
37
53
EN
Joe
Gildea
Senior Lecturer in Mathematics, Department of Mathematics
j.gildea@chester.ac.uk
In this paper, we investigate the Zassenhaus conjecture for $PSL(4,3)$ and $PSL(5,2)$. Consequently, we prove that the Prime graph question is true for both groups.
Zassenhaus Conjecture,torsion unit,partial augmentation,integral group ring
http://ijgt.ui.ac.ir/article_12010.html
http://ijgt.ui.ac.ir/article_12010_d29092385cbddb565eeadc78f175a1e0.pdf