فهرست مطالب

International Journal of Group Theory - Volume:6 Issue:1, 2016
  • Volume:6 Issue:1, 2016
  • تاریخ انتشار: 1395/10/05
  • تعداد عناوین: 6
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  • Seyed Ali Moosavi * Pages 1-7
    The concept of the bipartite divisor graph for integer subsets has been considered in [ýMý. ýAý. ýIranmanesh and Cý. ýEý. ýPraegerý, ýBipartite divisor graphs for integer subsetsý, ý{\em Graphs Combin.}ý, ý{\bf 26} (2010) 95--105ý.]ý. ýIn this paperý, ýwe will consider this graph for the set of character degrees of a finite group G and obtain some properties of this graphý. ýWe show that if G is a solvable groupý, ýthen the number of connected components of this graph is at most 2 and if G is a non-solvable groupý, ýthen it has at most 3 connected componentsý. ýWe also show thatý ýthe diameter of a connected bipartite divisor graph is bounded by 7 and obtain some properties of groups whose graphs are complete bipartite graphsý.
    Keywords: ?Bipartite divisor graph?, ?character degree?, ?connected component?, ?diameter
  • Laurent Poinsot* Pages 9-16
    This contribution mainly focuses on some aspects of Lipschitz groupsý, ýi.e.ý, ýmetrizable groups with Lipschitz multiplication and inversion mapý. ýIn the main result it is proved that metric groupsý, ýwith a translation-invariant metricý, ýmay be characterized as particular group objects in the category of metric spaces and Lipschitz mapsý. ýMoreoverý, ýup to an adjustment of the metricý, ýany metrizable abelian group also is shown to be a Lipschitz groupý. ýFinally 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−xý, ýin the setting of complete Lipschitz groupsý.
    Keywords: ýLipschitz mapsý, ýgroup object in a categoryý, ýtranslation, invariant metric
  • Leyli Jafari Taghvasani, Mohammad Zarrin * Pages 17-20
    For any group Gý, ýwe define an equivalence relation ∼ as belowý: ý
    ∀ gý,ýh∈G g∼h⟺|g|=|h|
    ý ýthe set of sizes of equivalence classes with respect to this relation is called the same-order type of G and denote by α(G)ý. ýIn this paperý, ýwe give a partial answer to a conjecture raised by Shený. ýIn factý, ýwe show that if G is a nilpotent groupý, ýthen |π(G)|≤|α(G)|ý, ýwhere π(G) is the set of prime divisors of order of Gý. ýAlso we investigate the groups all of whose proper subgroupsý, ýsay H have |α(H)|≤2ý.
    Keywords: ?Nilpotent groups?, ?Same, order type?, ?Schmidt group
  • Giovanni Vincenzi * Pages 21-27
    A subgroup X ofý ýa group G is said to be an {\it{\scriptsize\calligra H}ý -ýsubgroup} ifý ýNG(X)∩Xg≤X for each element g belonging to Gý. ýIn [Mý. ýBianchi and e.a.ý, ýOn finite soluble groups in which normality is a transitive relationý, ý{\em Jý. ýGroup Theory}ý, ý{\bf 3} (2000) 147--156.] the authors showed that finite groups in which every subgroup has the {\scriptsize\calligra H}ý -ýproperty are exactly soluble groups in which normality is a transitive relationý. ýHere we extend this characterization to groups without simple sectionsý.
    Keywords: ?{{scriptsizecalligra H}? ?, subgroups, ? ?{scriptsizecalligra T}? ?, groups, ? ?pronormal subgroups?, ?weakly normal subgroups?, ?pronorm, {scriptsizecalligra H}?, ?norm of a group.}
  • Marzieh Akbari, Alireza Moghaddamfar * Pages 29-35
    The noncommuting graph ∇(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.
    Keywords: nonabelian group, noncommuting graph, split graph
  • Joe Gildea * Pages 37-53
    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.
    Keywords: Zassenhaus Conjecture, torsion unit, partial augmentation, integral group ring