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A note on the power graph of a finite group | ||
| International Journal of Group Theory | ||
| مقاله 1، دوره 5، شماره 1، خرداد 2016، صفحه 1-10 اصل مقاله (254.85 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2016.6013 | ||
| نویسندگان | ||
| Zeinab Mehranian1؛ Ahmad Gholami1؛ Ali Reza Ashrafi* 2 | ||
| 1Univ Qom | ||
| 2University of Kashan | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Power graph؛ generalized join؛ automorphism group | ||
| مراجع | ||
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