13955

12682

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

۱۳۹۷

[يازده]، ۷۸ص.: مصور

عليرضا عبدالهي(دانشگاه اصفهان)

1397/07/17

كتابنامه

علوم رياضي

رياضي

ID12682

Conjugation action of finite groups with on sets of subgroups Azadeh Orak a orak@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 Mohsen Ghoraishi Associate Professor m ghoraishi@scu ac ir2000 MSC 20D10 20D20 20D25 20D35 20E34 20E45Keywords Finite group Noncyclic group Non normal subgroup Non subnormal subgroup Solvable group Abstract This M Sc thesis is based on the following papers Feng A and Liu Z Finite groups having exactly two conjugate classes of non subnormal subgroups Comm Algebra 43 9 2015 3840 3847 Lu J and Meng W On finite groups with non subnormal subgroups Comm Algebra 45 5 2017 2043 2046 Lu J and Meng W On solvability of finite groups with few non normal subgroups Comm Algebra 43 2015 1752 1756 Lu J Pang L and Qiu Y Finite groups with few non normal subgroups Journal of Algebra and Its Appli cations 14 4 2015 Mousavi H Groups with one conjugacy class of non normal subgroups A short proof Bull Iranian Math Soc 41 6 2015 1493 1495 For a finite group G let v G denote the number of conjugacy classes of non normal subgroups of G and vnc G denote the number of conjugacy classes of non normal noncyclic subgroups of G Every finite group G satisfyingv G 2 G or vnc G G is solvable and for a finite nonsolvable group G v G 2 G 1if and only if G A5 We show that if G is a finite group then v G 1 if and only if G Q P where Q Zq P Zpn and Mpn 1 where g h g pn hp 1 g h Q P 1 where p and q primes with p q 1 or G n 1 n 2 if p 2 and n 3 if p 2 g 1 pWe also show that if G is a finite nilpotent group such that v G G Then either G is a Dedekind group that is v G 0 or G is isomorphic to Mpm 1 or Mpm 1 Zq where p and q are distinct primes We show that if G is a finite non nilpotent group such that v G G then G 4 and calssify suchgroups For a finite group G let n G denotes the number of conjugacy classes of non subnormal subgroups of G We showthat if n G 2 G then G is solvable Also if G is non solvable then n G 2 G 1 if and only ifG A5 We show that if G is a finite group with n G 2 having a non normal maximal Sylow p subgroup then G a b c1 cn ap 1 cq b ap ci cj 1 i j 1 2 n m i cb ci 1 i 1 2 n 1 cb c11 cd2 cdn d i n n 2where f x xn dn xn 1 d2 x d1 is an irreducible polynomial over the field Fq dividing xp 1 and q n 1 mod p q is prime We classify finite groups G with n G 2 and G 2 3

بيژن طائري

محسن قريشي

عليرضا عبدالهي، محمدرضا ودادي