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چكيده انگليسي :

Groups whose prime graph on conjugacy class sizes has few complete vertices Seyed Meysam Hosseini Meysam Hosseini@math iut ac ir December 15 2014 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 MohammadReza Vedadi mrvedadi@cc iut ac ir2000 MSC 20E45 20D60Keywords Coprime action Nilpotent Solvable Prime graph Complete vertex Fitting height Abstract This M Sc thesis is based on the following paperCarlo Casolo Silvio Dolfi Emanuele Pacifici Lucia Sanus Groups whose prime graph on conjugacyclass sizes has few complete vertices Journal of Algebra 364 2012 1 12 Let G be a finite group and let cs G be the set of the sizes of the conjugacy classes of G we denote by G theprime graph built on cs G the vertices of G are the prime numbers dividing some element of cs G and two distinct vertices p q are adjacent in G if and only if there exists an element in cs G that is divisible by pq We write V G and E G for the sets of vertices and edges respectively of a prime graph G A vertex of a graph is said to be complete if it is adjacent to all other vertices of the graph In this thesis we considerthe situation when G has few complete vertices and our aim is to investigate the influence of this property onthe group structure of G More precisely we show G is a finite group are that at most one vertex of G that if iscomplete Then G is solvable and the Fitting height of G is at most 3 In fact under the assumptions of above fact the factor group G F G F G is nilpotent whence G is a nilpotentby nilpotent by abelian group if in addition the prime 2 is not a complete vertex then G F G F G turns out tobe abelian thus G is nilpotent by metabelian We show with an example the above description is from one point ofview the best possible in the sense that the Fitting series of G can have length 3 The conclusion of the above theorem is as follows Let G be a finite solvable group such that G is of boundedFitting height for conjugacy class sizes Then the Fitting height of G is at most 3 If we strengthen the hypothesisof above theorem and assuming that G has no complete vertices then we get a stronger conclusion Let G bea finite group Assume that no vertex of G is complete Then up to an abelian direct factor G KH withK G K and H are abelian groups of coprime order Moreover K G K Z G 1 and the prime divisorsof K respectively H are pairwise adjacent vertices in G There are two classes of groups that obviously satisfy the assumptions of above theorem One of them consists of thegroups G such that G has diameter 3 The other one consists of direct products of groups G such that G isdisconnected where the direct factors have pairwise coprime orders It might be reasonable to ask whether these two classes of groups contain essentially all the groups satisfying thehypotheses of above theorem We show that with an example this is false and if the hypotheses are strengthened and G is assumed to be a regular graph then both classes of groups satisfying the assumptions of theorem and othergroups not satisfying them This leads to the following theorem Let G be a finite group and assume that G is anoncomplete regular graph of degree d with n vertices Then G A G1 Gn 2m where m n 1 d the Gi are m balanced D groups of pairwise coprime orders and A is an abelian group Conversely for a group Gof this kind G is the join of n 2m copies of m

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گروه هايي كه گراف اول آن ها روي اندازه رده هاي مزدوجي تعداد كمي راس كامل دارند

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