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چكيده انگليسي :

On finite solvable groups whose cyclic p subgroups of equal order are conjugate Nahid Didban n didban@math iut ac ir June 22 2016 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Professor Bijan Taeri b taeri@cc iut ac irAdvisor Dr Atefeh Ghorbani a ghorbani@cc iut ac ir2000 MSC 20F15 20D05 20D10 Keywords Finite group Conjugacy class Solvable group Sylow subgroup Nilpotent group Suzuki group Abstract This M Sc thesis is based on the following paperSezgin Sezer On finite solvable groups whose cyclic p subgroups of equal order are conjugate Journal of Algebra415 2014 214 233 A finite group G is called a P group if every two cyclic subgroups of prime power order are conjugate in G inother words G is a P group if p is any prime divisor of G and pk G where k is a positive integer then theaction of G by conjugation on the set of cyclic subgroups order pk is transitive Some examples of P groups areA4 A 1 pd SL 2 3 P SL 2 7 In this thesis we study finite P groups We show that a homomorphicimage of a P group is again a P group Every nilpotent P group is cyclic We also show that if G is a solvable P group then G the derived subgroup of G is a Hall subgroup of G and G G is cyclic and every non cyclicelementary abelian Sylow subgroup of G lies in the Fitting subgroup of G We prove that if G is a P group and Nis a normal subgroup then G splits over N if and only if N is a Hall subgroup We show that a metacyclic group Gis a P group if and only if each Sylow subgroup is cyclic We show that most solvable P groups are very large subgroups of V A 1 pd where V GF p d is avector space of dimension d over the field GF p We clarify the structure of the Sylow p subgroups of a solvable P group where p is an odd prime Also we show that a Sylow 2 subgroup of a solvable P group only can cyclic or elementary or quaternion of order 8 or a Suzuki 2 group Finally we investigate solvable P subdirectly groupsG in two cases F G Fitting subgroup of G abelian and non abelian In particular we prove the following theorems Theorem A Let G be solvable group Then G is a P subdirectly irreducible group if and only if one of thefollowing holds 1 G A B where A GF p d is elementary abelian of order pd p B B being a P group whose Sylow subgroups are all cyclic acting transitively on the set of subgroups of A of order p 2 G A B where either A GF 5 2 and B SL 2 3 or A GF 11 2 and B SL 2 3 Ca with a 1 or a 5 where Ca is a cyclic group of order a In this case B acts transitively on the set of subgroups of A of order 5 and we see that G A F G and G 4 1 3 either G SL 2 3 or G is a P group whose Sylow 2 subgroups are Suzuki 2 groups Theorem B Let G be a group and Z G Assume the following hold a Z and G Z are both P groups and b for every cyclic p subgroup X of G either X is a subgroup of Z or otherwise X contains a Sylow p subgroup of Z Then G is a P group Theorem C Let G be a p solvable group where p is an odd prime Assume that every two cyclic p subgroupsof the same order are conjugate in G Then G has abelian Sylow p subgroups which are either cyclic or elementaryabelian In particular G has p lenght 1

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گروه هاي حل پذير متناهي كه p- زير گروه هاي دوري از مرتبه يكسان آن ها مزدوج هستند

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