2017-08-24T10:03:11Z
http://ijgt.ui.ac.ir/?_action=export&rf=summon&issue=3874
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2017
6
1
On bipartite divisor graph for character degrees
Seyed Ali
Moosavi
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
2017
03
01
1
7
http://ijgt.ui.ac.ir/article_9852_9330be05a79da3999795a5098e2d1f78.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2017
6
1
Lipschitz groups and Lipschitz maps
Laurent
Poinsot
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
2017
03
01
9
16
http://ijgt.ui.ac.ir/article_10506_0454ef2c3bfb84c29f58794c6552fea9.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2017
6
1
Shen's conjecture on groups with given same order type
Leyli
Jafari Taghvasani
Mohammad
Zarrin
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
2017
03
01
17
20
http://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2017
6
1
A characterization of soluble groups in which normality is a transitive relation
Giovanni
Vincenzi
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.}
2017
03
01
21
27
http://ijgt.ui.ac.ir/article_10890_1b9c272898954a2b55df94ab7233e6e9.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2017
6
1
Groups for which the noncommuting graph is a split graph
Marzieh
Akbari
Alireza
Moghaddamfar
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
2017
03
01
29
35
http://ijgt.ui.ac.ir/article_11161_c917794bead2c36c83dfeb54b217fb49.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2017
6
1
Torsion units for some projected special linear groups
Joe
Gildea
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
2017
03
01
37
53
http://ijgt.ui.ac.ir/article_12010_d29092385cbddb565eeadc78f175a1e0.pdf