2017
6
1
1
0
On bipartite divisor graph for character degrees
2
2
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) 95105.]. 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 nonsolvable 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.
1

1
7


Seyed Ali
Moosavi
University of Qom
University of Qom
Iran
s.a.mousavi@qom.ac.ir
bipartite divisor graph
character degree
connected component
diameter
Lipschitz groups and Lipschitz maps
2
2
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 translationinvariant 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 $1x$, in the setting of complete Lipschitz groups.
1

9
16


Laurent
Poinsot
University Paris 13, Paris Sorbonne Cité
University Paris 13, Paris Sorbonne Cité
France
laurent.poinsot@lipn.univparis13.fr
Lipschitz maps
group object in a category
translationinvariant metric
Shen's conjecture on groups with given same order type
2
2
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 sameorder 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 $alpha{(H)}leq 2$.
1

17
20


Leyli
Jafari Taghvasani
Department of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, Iran
Department of Mathematics, University of
Iran
l.jafari@sci.uok.ac.ir


Mohammad
Zarrin
University of Kurdistan
University of Kurdistan
Iran
zarrin@ipm.ir
Nilpotent groups
Sameorder type
Schmidt group
A characterization of soluble groups in which normality is a transitive relation
2
2
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) 147156.] 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.
1

21
27


Giovanni
Vincenzi
University of Salerno
University of Salerno
Italy
vincenzi@unisa.it
{{scriptsizecalligra H} subgroups
{scriptsizecalligra T} groups
pronormal subgroups
weakly normal subgroups
pronorm and {scriptsizecalligra H} norm of a group.}
Groups for which the noncommuting graph is a split graph
2
2
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.
1

29
35


Marzieh
Akbari
K. N. Toosi University of Technology
K. N. Toosi University of Technology
Iran
m.akbari@dena.kntu.ac.ir


Alireza
Moghaddamfar
K.N. Toosi University of Technology
K.N. Toosi University of Technology
Iran
moghadam@kntu.ac.ir
nonabelian group
noncommuting graph
split graph
Torsion units for some projected special linear groups
2
2
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.
1

37
53


Joe
Gildea
Senior Lecturer in Mathematics, Department of Mathematics
Senior Lecturer in Mathematics, Department
United Kingdom
j.gildea@chester.ac.uk
Zassenhaus Conjecture
torsion unit
partial augmentation
integral group ring