On bipartite divisor graph for character degrees
Seyed Ali
Moosavi
University of Qom
author
text
article
2017
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
6
v.
1
no.
2017
1
7
http://ijgt.ui.ac.ir/article_9852_9330be05a79da3999795a5098e2d1f78.pdf
dx.doi.org/10.22108/ijgt.2017.9852
Lipschitz groups and Lipschitz maps
Laurent
Poinsot
University Paris 13, Paris Sorbonne Cité
author
text
article
2017
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
6
v.
1
no.
2017
9
16
http://ijgt.ui.ac.ir/article_10506_0454ef2c3bfb84c29f58794c6552fea9.pdf
dx.doi.org/10.22108/ijgt.2017.10506
Shen's conjecture on groups with given same order type
Leyli
Jafari Taghvasani
Department of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, Iran
author
Mohammad
Zarrin
University of Kurdistan
author
text
article
2017
eng
For any group $G$, we define an equivalence relation $\thicksim$ as below: \[\forall \ g, h \in G \ \ g\thicksim h \Longleftrightarrow |g|=|h|\] 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 $|\alpha{(H)}|\leq 2$.
International Journal of Group Theory
University of Isfahan
2251-7650
6
v.
1
no.
2017
17
20
http://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdf
dx.doi.org/10.22108/ijgt.2017.10631
A characterization of soluble groups in which normality is a transitive relation
Giovanni
Vincenzi
University of Salerno
author
text
article
2017
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
6
v.
1
no.
2017
21
27
http://ijgt.ui.ac.ir/article_10890_1b9c272898954a2b55df94ab7233e6e9.pdf
dx.doi.org/10.22108/ijgt.2017.10890
Groups for which the noncommuting graph is a split graph
Marzieh
Akbari
K. N. Toosi University of Technology
author
Alireza
Moghaddamfar
K.N. Toosi University of Technology
author
text
article
2017
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
6
v.
1
no.
2017
29
35
http://ijgt.ui.ac.ir/article_11161_4a0587eb7f156827981f201aed7d43c2.pdf
dx.doi.org/10.22108/ijgt.2017.11161
Torsion units for some projected special linear groups
Joe
Gildea
Senior Lecturer in Mathematics, Department of Mathematics
author
text
article
2017
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
6
v.
1
no.
2017
37
53
http://ijgt.ui.ac.ir/article_12010_d29092385cbddb565eeadc78f175a1e0.pdf
dx.doi.org/10.22108/ijgt.2017.12010