University of IsfahanInternational Journal of Group Theory2251-76506120170301On bipartite divisor graph for character degrees17985210.22108/ijgt.2017.9852ENSeyed AliMoosaviUniversity of QomJournal Article20150416The 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.</em>, <strong>26</strong> (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.https://ijgt.ui.ac.ir/article_9852_9330be05a79da3999795a5098e2d1f78.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76506120170301Lipschitz groups and Lipschitz maps9161050610.22108/ijgt.2017.10506ENLaurentPoinsotUniversity Paris 13, Paris Sorbonne CitéJournal Article20130408This 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.https://ijgt.ui.ac.ir/article_10506_0454ef2c3bfb84c29f58794c6552fea9.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76506120170301Shen's conjecture on groups with given same order type17201063110.22108/ijgt.2017.10631ENLeyliJafari TaghvasaniDepartment of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, IranMohammadZarrinUniversity of KurdistanJournal Article20150609For 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 $|\alpha{(H)}|\leq 2$.https://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76506120170301A characterization of soluble groups in which normality is a transitive relation21271089010.22108/ijgt.2017.10890ENGiovanniVincenziUniversity of SalernoJournal Article20150709A subgroup $X$ of a group $G$ is said to be an <em>H</em>-subgroup if <em>N<sub>G</sub>(X) ∩ X<sup>g </sup>≤ X</em> 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</em>, <strong> 3</strong> (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.https://ijgt.ui.ac.ir/article_10890_1b9c272898954a2b55df94ab7233e6e9.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76506120170301Groups for which the noncommuting graph is a split graph29351116110.22108/ijgt.2017.11161ENMarziehAkbariK. N. Toosi University of TechnologyAlirezaMoghaddamfarK.N. Toosi University of TechnologyJournal Article20150610The 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<br /> 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.https://ijgt.ui.ac.ir/article_11161_4a0587eb7f156827981f201aed7d43c2.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76506120170301Torsion units for some projected special linear groups37531201010.22108/ijgt.2017.12010ENJoeGildeaSenior Lecturer in Mathematics, Department of MathematicsJournal Article20151125In this paper, we investigate the Zassenhaus conjecture for $PSL(4,3)$ and $PSL(5,2)$. Consequently,<br /> we prove that the Prime graph question is true for both groups.https://ijgt.ui.ac.ir/article_12010_d29092385cbddb565eeadc78f175a1e0.pdf