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چكيده انگليسي :

The generating graph of some monolithic groups Mohsen Ahmadi mohsen ahmadi@math iut ac ir August 19 2015 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Bijan Taeri b taeri@cc iut ac irAdvisor Dr Seyed Ghahreman Taherian taherian@cc iut ac ir2000 MSC 20B25 05C25Keywords Generating graph Hamiltonian cycles Monolithic groups Abstract This M Sc thesis is based on the following paperE Crestani A Lucchini The generating graph of some monolithic groups Journal of Combinatorial Theory SeriesA 119 2012 615 626 In this thesis we deal with wreath product of finite simple groups and cyclic groups We investigate the basic prop erties of wreath products Also we present classical simple groups and describe monolithic groups We calculate thegenerators of S Cm the wreath product of S by Cm where S is a simple finite non abelian group For a finite group G let G denote the graph defined on the nonidentity elements of G in such a way that twodistinct vertices are connected by an edge if and only if they generate G In this thesis we study some propertiesthe generating graph of some monolithic groups We prove that for a monolithic group G with non abelian socle Nsuch that G N is a cyclic group if G contains a Hamiltonian cycle then G is pancyclic provided that N is large enough Also we prove that if m is odd and the number of prime numbers dividing m is at most 140 thenthere exists a positive integer such that if S is a simple group of Lie type and S then the graph S Cm contains a Hamiltonian cycle Let be a graph with n vertices and let d denote the degree of the vertex The closure cl of is the graph on the same set of vertices constructed from by adding the new edge u for every pair of non adjacent verticesu and such that d u d n Let m be a natural number We will denote by m the m 1 closureof the generating graph Cm of the cyclic group of order m Let Um be the set of vertices of m correspondingto the elements of the subset g Cm m g is odd We will say that m is Hamiltonian if any u Um isconnected in m to any other vertex Suppose G S Cm the wreath product of S by Cm where S is a simplefinite non abelian group In this thesis we study a certain conditions in which G consist of a Hamiltonian cycleand prove the following results 1 Assume that m is Hamiltonian There exists a positive integer m which depends on m such that if S is a non abelian simple group with S then the graph S Cm contains a Hamiltonian cycle s 2 Let m i 1 pi i where ps ps 1 p1 are distinct primes and i 0 for every 1 i s Assume that one of the following conditions is satisfied m a m is odd and m 1 where m denotes the Euler function 6 m 1 1 3A 2 1 1 where A b 6 p2 1 i s 3 m i Then there exists a positive integer such that if S is a simple group of Lie type and S then the graph S Cm contains a Hamiltonian cycle

احمدي، محسن

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