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AIMS Mathematics

2023, Volume 8, Issue 12: 30981-30991. doi: 10.3934/math.20231585
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Research article

On the structure of finite groups associated to regular non-centralizer graphs

  • Department of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi Arabia
  • Received: 16 September 2023 Revised: 07 November 2023 Accepted: 13 November 2023 Published: 20 November 2023

  • MSC : 05C25, 20B05

  • The non-centralizer graph of a finite group $ G $ is the simple graph $ \Upsilon_G $ whose vertices are the elements of $ G $ with two vertices are adjacent if their centralizers are distinct. The induced non-centralizer graph of $ G $ is the induced subgraph of $ \Upsilon_G $ on $ G\setminus Z(G) $. A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups and induced regular groups. We prove that if a group $ G $ is regular, then $ G/Z(G) $ as an elementary $ 2 $-group. Using the concept of maximal centralizers, we succeeded in proving that if $ G $ is induced regular, then $ G/Z(G) $ is a $ p $-group. We also show that a group $ G $ is induced regular if and only if it is the direct product of an induced regular $ p $-group and an abelian group.

    Citation: Tariq A. Alraqad, Hicham Saber. On the structure of finite groups associated to regular non-centralizer graphs[J]. AIMS Mathematics, 2023, 8(12): 30981-30991. doi: 10.3934/math.20231585

    Related Papers:

  • The non-centralizer graph of a finite group $ G $ is the simple graph $ \Upsilon_G $ whose vertices are the elements of $ G $ with two vertices are adjacent if their centralizers are distinct. The induced non-centralizer graph of $ G $ is the induced subgraph of $ \Upsilon_G $ on $ G\setminus Z(G) $. A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups and induced regular groups. We prove that if a group $ G $ is regular, then $ G/Z(G) $ as an elementary $ 2 $-group. Using the concept of maximal centralizers, we succeeded in proving that if $ G $ is induced regular, then $ G/Z(G) $ is a $ p $-group. We also show that a group $ G $ is induced regular if and only if it is the direct product of an induced regular $ p $-group and an abelian group.


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