A note on the power graph of a finite group Zeinab Mehranian Univ Qom author Ahmad Gholami Univ Qom author Ali Reza Ashrafi University of Kashan author text article 2016 eng ‎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. International Journal of Group Theory University of Isfahan 2251-7650 5 v. 1 no. 2016 1 10 http://ijgt.ui.ac.ir/article_6013_2f59939f01f22a9a5a68b464a1d66678.pdf dx.doi.org/10.22108/ijgt.2016.6013 Finite simple groups which are the products of symmetric or alternating groups with $L_{3}(4)$ Gholamreza Rezaeezadeh University of Shahrekord author Mohammad Reza Darafsheh University of Tehran author Ebrahim Mirdamadi University of Shahrekord author text article 2016 eng In this paper‎, ‎we determine the simple groups $G=AB$‎, ‎where $B$ is isomorphic to $L_{3}(4)$ and $A$ isomorphic to an alternating or a symmetric group on $n\geq5$‎, ‎letters‎. International Journal of Group Theory University of Isfahan 2251-7650 5 v. 1 no. 2016 11 16 http://ijgt.ui.ac.ir/article_5505_e4c536aa978082836092658f64f08221.pdf dx.doi.org/10.22108/ijgt.2016.5505 Characterization of projective general linear groups Alireza Khalili Asboei Farhangian University, Shariati Mazandaran, Iran author text article 2016 eng ‎Let $G$ be a finite group and $\pi_{e}(G)$ be the set of element orders of $G$‎. ‎Let $k \in \pi_{e}(G)$ and $s_{k}$ be the number of elements of order $‎k$ in $G$‎. ‎Set nse($G$):=$\{ s_{k} | k \in \pi_{e}(G)\}$‎. ‎In this paper‎, ‎it‎ ‎is proved if $|G|=|$ PGL$_{2}(q)|$‎, ‎where $q$ is odd prime power and nse$‎(G)=$nse$($PGL$_{2}(q))$‎, ‎then $G \cong$PGL$_{2}(q)$‎. International Journal of Group Theory University of Isfahan 2251-7650 5 v. 1 no. 2016 17 28 http://ijgt.ui.ac.ir/article_5634_ec32a14b9a19c927c5f00f5fa7454057.pdf dx.doi.org/10.22108/ijgt.2016.5634 A characterization of $\mathbf{L_2(81)}$ by nse Leila Mousavi Department of Mathematical Sciences; Isfahan University of Technology; Isfahan 84156-83111; Iran. author Bijan Taeri Department of Mathematical Sciences;, Isfahan University of Technology author text article 2016 eng Let $\pi_e(G)$ be the set of element orders of a finite group $G$‎. ‎Let $nse(G)=\{m_n\mid n\in\pi_e(G)\}$‎, ‎where $m_n$ be the number of elements of order $n$ in $G$‎. ‎In this paper‎, ‎we prove that if $nse(G)=nse(L_2(81))$‎, ‎then $G\cong L_2(81)$‎. International Journal of Group Theory University of Isfahan 2251-7650 5 v. 1 no. 2016 29 35 http://ijgt.ui.ac.ir/article_5843_2a42ff7ced0280cc4820db4840f4709b.pdf dx.doi.org/10.22108/ijgt.2016.5843 A note on the affine subgroup of the symplectic group Jamshid Moori North-West University (Mafikeng) P Bag X2046, Mmabatho 2735, South Africa author Bernardo Rodrigues University of KwaZulu-Natal Durban, South Africa author text article 2016 eng ‎We examine the symplectic group $Sp_{2m}(q)$ and its corresponding affine subgroup‎. ‎We construct the affine subgroup and show that it is a split extension‎. ‎As an illustration of the above we study the affine subgroup $2^5{:}Sp_4(2)$ of the group $Sp_6(2)$‎. International Journal of Group Theory University of Isfahan 2251-7650 5 v. 1 no. 2016 37 51 http://ijgt.ui.ac.ir/article_7281_ac143a61d9651c37ed5af49f57a3661d.pdf dx.doi.org/10.22108/ijgt.2016.7281