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12488

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

۱۳۹۷

هشت، ۸۳ص.: مصور

عليرضا عبداللهي استاد داور پايان نامه از دانشگاه اصفهان است.

بيژن طائري

محمدرضا ودادي

انگليسي به فارسي; فارسي به انگليسي

عليرضا عبداللهي، محمود بهبودي

1397/05/23

كتابنامه

علوم رياضي

رياضي

ID12488

Finite Groups with few TI subgroups Farzad Mazrooei Sebdani f mazrooei@math iut ac ir June 13 2018 Master of Science Thesis Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Bijan Taeri b taeri@cc iut ac irAdvisor Dr Mohammad Reza Vedadi mrvedadi@cc iut ac ir2000 MSC 20D10 20D20 20D60Keywords TI subgroup Metacyclic subgroup QH group QTI subgroup CTI group TI group ATI group Abstract This M Sc thesis is based on the following papers Li Sh Shen Zh Ni Du Finite Groups with few TI subgroups Communications in Algebra 43 7 2680 2689 2015 Abdollahi A Mousavi H Finite Nilpotent Groups whose Cyclic subgroups are TI subgroups Bull Malays Sci Soc 2015 A subgroup H of a finite group G is called a TI subgroup if H H x 1 or H H x H for all x G A groupG is called a TI group if all of whose subgroups are TI subgroups The concept of TI subgroup is fundamentalrole in the study of finite groups In this thesis we consider QH groups i e groups whose all metacyclic subgroupsare TI subgroups We show that all QH groups are solvable and classify QH groups completely Moreover if thecondition of metacyclic is replaced by the conditions of cyclic or abelian then such group may be non solvable Forthe p groups we have the following important theorem Theorem Let G be a finite p group p a prime Then G is a QH group if and only if G is one of the followinggroups 1 An abelian p group n p a b apn bp 1 ab a1 pn 1 for some n 2 2 M p n n 1 a b c a c b c 1 for some n 1 M p p pn ap bp cp cp 1 a b 3 4 A Hamiltonian 2 group 5 Q8 D8 the central product of Q8 the quaternion group of order 8 and D8 the dihedral group of order 8 We also have the following theorem on QH groups Theorem Let G be a finite group all of whose metacyclic subgroup are TI subgroups Then G is solvable and oneof the following statements holds 1 G is a p group as the list of upper theorem 2 G is a Dedekind group 3 G is a Frobenius group with kernel K and with complement H where K is abelian and H is cyclic A group G is called a CTI group if any cyclic subgroup of G is a TI subgroup A characterization of finite nilpotent CTI groups is given in the following theorem Theorem Let G be a finite nilpotent group then G is a CTI group if and only if one of the following holds 1 G is Dedekindian i e all subgroups of G are normal in G 2 G is a p group of exponent p for some prime p 3 G is a p group for some prime p such that G 1 Gp is of order p and G Gp is a central cyclicsubgroup of G

بيژن طائري

محمدرضا ودادي

عليرضا عبداللهي، محمود بهبودي