A note on the power graph of a finite group

Document Type : Research Paper

Authors

1 Univ Qom

2 University of Kashan

Abstract

‎Suppose $\Gamma$ is a graph with $V(\Gamma) = \{ 1‎, ‎2,\dots‎, ‎p\}$‎ ‎and $ \mathcal{F} = \{\Gamma_1,\dots‎, ‎\Gamma_p\} $ is a family of‎ ‎graphs such that $n_j = |V(\Gamma_j)|$‎, ‎$1 \leq j \leq p$‎. ‎Define‎ ‎$\Lambda = \Gamma[\Gamma_1,\dots‎, ‎\Gamma_p]$ to be a graph with‎ ‎vertex set $ V(\Lambda)=\bigcup_{j=1}^pV(\Gamma_j)$ and edge set‎ ‎$E(\Lambda)=\big(\bigcup_{j=1}^pE(\Gamma_j)\big)\cup\big(\bigcup_{ij\in‎ ‎E(\Gamma)}\{uv;u\in V(\Gamma_i),v\in V(\Gamma_j)\}\big) $‎. ‎The‎ ‎graph $ \Lambda$ is called the $\Gamma$-join of $ \mathcal{F}$‎. ‎The power graph $\mathcal{P}(G)$ of a group $G$ is the graph‎ ‎which has the group elements as vertex set and two elements are‎ ‎adjacent if one is a power of the other‎. ‎The aim of this paper is‎ ‎to prove that $\mathcal{P}(\mathbb{Z}_{n}) = K_{\phi(n)+1}‎ + ‎\Delta_n[K_{\phi(d_1)}‎, ‎K_{\phi(d_2)},\dots‎, ‎K_{\phi(d_{p})}]$‎, ‎where $\Delta_n$ is a graph with vertex and edge sets‎ ‎$V(\Delta_n)=\{d_i \ | \ 1,n\not = d_i | n‎, ‎1\leq i\leq p\}$ and‎ ‎$ E(\Delta_n)=\{ d_id_j \ | \ d_i|d_j‎, ‎1\leq i<j\leq p\}$‎, ‎respectively‎. ‎As a consequence it is proved that‎ ‎$Aut(\mathcal{P}(\mathbb{Z}_{n}))\cong‎ ‎S_{\phi(n)+1}\times\prod_{1,n\not=d|n}S_{\phi(d)}.$ This proves a‎ ‎recent conjecture by Doostabadi et al‎. ‎[A‎. ‎Doostabadi‎, ‎A‎. ‎Erfanian and A‎. ‎Jafarzadeh‎, ‎Some results on the power graph of groups, ‎The Extended Abstracts of the 44th Annual Iranian Mathematics Conference‎, ‎27-30 August 2013‎, ‎Ferdowsi University of Mashhad‎, ‎Iran]‎. ‎Finally‎, ‎we‎ ‎apply our results to obtain complete descriptions of the power‎ ‎graphs of some finite groups‎.

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References

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History

  • Receive Date: 24 July 2014
  • Revise Date: 06 August 2014
  • Accept Date: 07 August 2014
  • Published Online: 01 March 2016

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