ORIGINAL_ARTICLE A note on the power graph of a finite group ‎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. http://ijgt.ui.ac.ir/article_6013_2f59939f01f22a9a5a68b464a1d66678.pdf 2016-03-01T11:23:20 2019-05-24T11:23:20 1 10 10.22108/ijgt.2016.6013 Power graph generalized join automorphism group Zeinab Mehranian mehranian.z@gmail.com true 1 Univ Qom Univ Qom Univ Qom AUTHOR Ahmad Gholami a.gholami@kashanu.ac.ir true 2 Univ Qom Univ Qom Univ Qom AUTHOR Ali Reza Ashrafi ashrafi@kashanu.ac.ir true 3 University of Kashan University of Kashan University of Kashan LEAD_AUTHOR
ORIGINAL_ARTICLE Finite simple groups which are the products of symmetric or alternating groups with $L_{3}(4)$ 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. http://ijgt.ui.ac.ir/article_5505_e4c536aa978082836092658f64f08221.pdf 2016-03-01T11:23:20 2019-05-24T11:23:20 11 16 10.22108/ijgt.2016.5505 Factorization product of groups symmetric groups Gholamreza Rezaeezadeh rezaeezadeh@sci.sku.ac.ir true 1 University of Shahrekord University of Shahrekord University of Shahrekord LEAD_AUTHOR Mohammad Reza Darafsheh darafsheh@ut.ac.ir true 2 University of Tehran University of Tehran University of Tehran AUTHOR Ebrahim Mirdamadi ebrahimmirdamadi@stu.sku.ac.ir true 3 University of Shahrekord University of Shahrekord University of Shahrekord AUTHOR  J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985. 1  M. R. Darafsheh and A. Mahmiani, Products of the symmetric or alternating groups with L3(3), Quasigroups 2 Related Systems, 17 (2009) 9–16. 3  M. R. Darafsheh, G. R. Rezaeezadeh and G. L. Walls, Groups which are the product of S6 and a simple group, Algebra Colloq., 10 (2003) 195–204. 4  J. D. Dixon and B. Mortimer, Permutation Groups, Springer-Verlag, New York, 1996. 5  M. Giudici, Factorization of sporadic simple groups, J. Algebra, 303 (2006) 311–323. 6  P. Kleidman and M. Liebeck, The struture of the finite classical groups, Cambridge University Press, 1990. 7  M. W. Liebeck, C. E. Praeger and J. Saxl, Regular subgroups of primitive permutation groups, Mem. Amer. Math. Soc., 203 (2010). 8  M. W. Liebeck, C. E. Praeger and J. Saxl, The maximal factorizations of the finite simple groups and their auto-morphism groups, Mem. Amer. Math. Soc., 86 1990. 9  W. R. Scott, Product of A5 and a finite simple group, J. Algebra, 37 (1975) 165–171. 10
ORIGINAL_ARTICLE Characterization of projective general linear groups ‎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$_{2}(q)$‎. http://ijgt.ui.ac.ir/article_5634_ec32a14b9a19c927c5f00f5fa7454057.pdf 2016-03-01T11:23:20 2019-05-24T11:23:20 17 28 10.22108/ijgt.2016.5634 ‎Element order‎ ‎set of the numbers of elements of the same order‎ ‎projective general linear group Alireza Khalili Asboei khaliliasbo@yahoo.com true 1 Farhangian University, Shariati Mazandaran, Iran Farhangian University, Shariati Mazandaran, Iran Farhangian University, Shariati Mazandaran, Iran LEAD_AUTHOR
ORIGINAL_ARTICLE A characterization of $\mathbf{L_2(81)}$ by nse 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 $G\cong L_2(81)$‎. http://ijgt.ui.ac.ir/article_5843_2a42ff7ced0280cc4820db4840f4709b.pdf 2016-03-01T11:23:20 2019-05-24T11:23:20 29 35 10.22108/ijgt.2016.5843 ‎set of the numbers of elements of the same order‎ ‎order of an element‎ ‎Thompson's problem Leila Mousavi l.mousavi@math.iut.ac.ir true 1 Department of Mathematical Sciences; Isfahan University of Technology; Isfahan 84156-83111; Iran. Department of Mathematical Sciences; Isfahan University of Technology; Isfahan 84156-83111; Iran. Department of Mathematical Sciences; Isfahan University of Technology; Isfahan 84156-83111; Iran. LEAD_AUTHOR Bijan Taeri b.taeri@cc.iut.ac.ir true 2 Department of Mathematical Sciences;, Isfahan University of Technology Department of Mathematical Sciences;, Isfahan University of Technology Department of Mathematical Sciences;, Isfahan University of Technology AUTHOR
ORIGINAL_ARTICLE A note on the affine subgroup of the symplectic group ‎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 $Sp_6(2)$‎. http://ijgt.ui.ac.ir/article_7281_ac143a61d9651c37ed5af49f57a3661d.pdf 2016-03-01T11:23:20 2019-05-24T11:23:20 37 51 10.22108/ijgt.2016.7281 Group extensions character table symplectic groups Jamshid Moori jamshid.moori@nwu.ac.za true 1 North-West University (Mafikeng) P Bag X2046, Mmabatho 2735, South Africa North-West University (Mafikeng) P Bag X2046, Mmabatho 2735, South Africa North-West University (Mafikeng) P Bag X2046, Mmabatho 2735, South Africa AUTHOR Bernardo Rodrigues rodrigues@ukzn.ac.za true 2 University of KwaZulu-Natal Durban, South Africa University of KwaZulu-Natal Durban, South Africa University of KwaZulu-Natal Durban, South Africa LEAD_AUTHOR  F. Ali, Fischer-Clifford Theory for Split and Non-Split Group Extensions, Ph. D. thesis, University of Natal, Pieter-maritzburg, 2001. 1  A. Basheer and J. Moori, On the non-split extension group 26·Sp(6,2), Bull. Iranian Math. Soc., 39 (2013) 1189–1212. 2  J. H. Conway, R.T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, 1985. 3  R. Gow, Some characters of affine subgroups of classical groups, J. London Math. Soc. (2), 2 (1976) 231–236. 4  The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.4. 5  J. Moori and T. Seretlo, On the Fischer-Clifford matrices of a maximal subgroup of the Lyons group Ly, Bul. of Iranian Math Soc., 39 (2013) 1037-1052. 6  Z. E. Mpono, Fischer Clifford Theory and Character Tables of Group Extensions, Ph. D. thesis, University of Natal, Pietermaritzburg, 1998. 7  B. G. Rodrigues, On the Theory and Examples of Group Extensions, Master’s thesis, University of Natal, Pieter-maritzburg, 1999. 8  J. J. Rotman, An Introduction to the Theory of Groups, Fourth ed., Springer-Verlag, New York, Inc., 1995. 9