13733

12483

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

۱۳۹۷

[ده]، ۹۴ص.: مصور

1397/05/22

كتابنامه

علوم رياضي

رياضي

ID12483

Groups with few non nilpotent subgroups Anahita Khadem Agha a khadem@math iut ac ir June 26 2018 Master of Science Thesis Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Bijan Taeri Professor b taeri@cc iut ac irAdvisor Dr Mahmood Behboodi Associate Professor mbehbood@cc iut ac ir2000 MSC 20D10 20D20 20E45 20D15 20F16 20F19Keywords Non nilpotent subgroups Non normal subgroups Solvable groups Minimal non nilpotent groups Abstract This M Sc thesis is based on the following papers Lu J K and Meng W On finite groups with non nilpotent subgroups Monatshefte f r Mathematik 179 1 2016 99 103 Brandl R Groups with few non nilpotent subgroups J Algebra Appl 16 2017 1750188 Let G be a finite group Let l G be the number of conjugacy classes of non normal non nilpotent subgroups ofG l0 G be the number of classes of non normal non nilpotent subgroups of the same order of G and G bethe number of conjugacy classes of non nilpotent subgroups of G A non nilpotent group G is called a minimalnon nilpotent if every proper subgroup of G is nilpotent In this thesis we show that every finite group G satisfyingl G G where G is the set of all primes dividing G is solvable and for a finite non solvable groupG l G G if and only if G A5 or SL 2 5 Then we classify all finite groups G with G 3 We have the following important theorem Theorem Let G be a finite group with G 2 and G 2 Then G is q nilpotent for some prime q SoG P Q where P Op G for some prime p q and the Sylow q subgroup Q x of G is cyclic Let QQ Then one of the following holds CQ P 1 Q q 2 and P x is minimal non nilpotent 2 Q q and either G H Zp or G H Zp2 is a central product where H is minimal non nilpotent P 3 Q q and and P are irreducible and nontrivial as Q modules In particular the nilpotency class P of P is at most 2 If P is abelian then it is homocyclic of exponent p2 We also have the following theorem for G 3 Theorem Let G 3 and G p q r Then either G S R where S is a minimal non nilpotent p q group and R Zr2 is cyclic or G P Q R where Q R is cyclic of order qr and P Q and P R are minimal non nilpotent or G P Q R where both P R and Q R are minimal non nilpotent and P and Qare elementary abelian P Q and R are Sylow p q and r subgroups of G and the primes are arranged suitably Finally we determine the structure of finite groups G with G 3 which are not r nilpotent for any prime r Theorem Let G 3 and assume that G has no normal r complement for any prime r G p q Assume that p q and set U Op G and K U Then 1 K 1 or p In any case U K is a chief factor of G and K Z G If K 1 then U is extra special 2 A Sylow q subgroup Q of G is cyclic of prime order q and Q acts irreducibly on U K Moreover NG Q Kis the Frobenius group of order pq In particular p divides q 1

بيژن طائري

محمود بهبودي

عليرضا عبداللهي، محمدرضا ودادي