eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2017-03-01
6
1
1
7
10.22108/ijgt.2017.9852
9852
On bipartite divisor graph for character degrees
Seyed Ali Moosavi
s.a.mousavi@qom.ac.ir
1
University of Qom
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, Graphs Combin., 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.
http://ijgt.ui.ac.ir/article_9852_9330be05a79da3999795a5098e2d1f78.pdf
bipartite divisor graph
character degree
connected component
diameter
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2017-03-01
6
1
9
16
10.22108/ijgt.2017.10506
10506
Lipschitz groups and Lipschitz maps
Laurent Poinsot
laurent.poinsot@lipn.univ-paris13.fr
1
University Paris 13, Paris Sorbonne Cité
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.
http://ijgt.ui.ac.ir/article_10506_0454ef2c3bfb84c29f58794c6552fea9.pdf
Lipschitz maps
group object in a category
translation-invariant metric
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2017-03-01
6
1
17
20
10.22108/ijgt.2017.10631
10631
Shen's conjecture on groups with given same order type
Leyli Jafari Taghvasani
l.jafari@sci.uok.ac.ir
1
Mohammad Zarrin
zarrin@ipm.ir
2
Department of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, Iran
University of Kurdistan
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 $|alpha{(H)}|leq 2$.
http://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdf
Nilpotent groups
Same-order type
Schmidt group
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2017-03-01
6
1
21
27
10.22108/ijgt.2017.10890
10890
A characterization of soluble groups in which normality is a transitive relation
Giovanni Vincenzi
vincenzi@unisa.it
1
University of Salerno
A subgroup $X$ of a group $G$ is said to be an H-subgroup if NG(X) ∩ Xg ≤ 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, J. Group Theory, 3 (2000) 147--156.] the authors showed that finite groups in which every subgroup has the H-property are exactly soluble groups in which normality is a transitive relation. Here we extend this characterization to groups without simple sections.
http://ijgt.ui.ac.ir/article_10890_1b9c272898954a2b55df94ab7233e6e9.pdf
$H$-subgroups
$T$-groups
pronormal subgroups
weakly normal subgroups
pronorm and H-norm of a group
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2017-03-01
6
1
29
35
10.22108/ijgt.2017.11161
11161
Groups for which the noncommuting graph is a split graph
Marzieh Akbari
m.akbari@dena.kntu.ac.ir
1
Alireza Moghaddamfar
moghadam@kntu.ac.ir
2
K. N. Toosi University of Technology
K.N. Toosi University of Technology
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.
http://ijgt.ui.ac.ir/article_11161_4a0587eb7f156827981f201aed7d43c2.pdf
nonabelian group
noncommuting graph
split graph
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2017-03-01
6
1
37
53
10.22108/ijgt.2017.12010
12010
Torsion units for some projected special linear groups
Joe Gildea
j.gildea@chester.ac.uk
1
Senior Lecturer in Mathematics, Department of Mathematics
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.
http://ijgt.ui.ac.ir/article_12010_d29092385cbddb565eeadc78f175a1e0.pdf
Zassenhaus Conjecture
torsion unit
partial augmentation
integral group ring