6850

6392

كارشناسي ارشد

رياض محض ﴿جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1390

[هفت]، 96ص.: مصور، جدول

ص.ع.به فارسي و انگليسي

بيژن طائري

محمد مشكوري

20/4/91

عليرضا عبدالهي، عاطفه قرباني

1396/09/14

كتابنامه

علوم رياضي

رياضي

ID6392

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Finite Nonsolvable Groups whose Character Graphs have no Triangles Marzieh Aslanifar m aslanifar@math iut ac ir 28 February 2012 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 Mohammad Mashkoori mohammad@cc iut ac ir2000 MSC 20C15 05C25Key Words Finite groups Nonsolvable groups Representation of groups Character of groups Nonlilear irreducible character Alternating groups Classical groups of Lie type Sporadic groups Character graph of a group Abstract This Msc thesis is based on the following paperTianze Li Yanjun Liu Xuling Song Finite nonsolvable groups whose character graphshave no triangles Journal of Algebra 323 2010 2290 2300 Let G be a nite group Then a representation of G is homomorphism X G GLn C forsome integer n The character of G a orded by representation X is a function G C suchthat g tr X g for all g G Let and be characters of G The inner product of and 1 is de ned as g g The character is irreducible if and only if 1 G g GThe number n 1 called the degree of The character is called a linear character if 1 1 otherwise it is called nonlinear There are several kind of graphs attached to nite groups such as conjugacy graphs degreegraphs and character graphs In this thesis we shall focus on character graphs The charactergraph G of a nite group G is de ned in the following way The vertices of the graph are thenonlinear complex irreducible characters of G and there is an edge between two vertices and if and only if 1 and 1 have a common prime divisor Let G be a nite group Irr G isthe set of its complex irreducible character and Irr1 G is the subset of Irr G consisting of thenonlinear ones If G is an abelian group then the graph of G is empty If G is nonabelian according to theclassi cation theorem of nite simple groups G is isomorphism to one of the following An forn 5 a classical group of Lie type i e linear groups unitary groups symplectic groups andorthogonal groups or sporadic simple groups In this thesis we shall show that A5 is the onlyexception for nonsolvable groups The following theorem is the main result Let G be a nitenonsolvable group whose character graph has no triangles Then G A5 the alternating group on ve letters According to the above theorem we can get that for a nite group its character graph hasno triangles if and only if the graph dose not contain a cycle i e each connected componentof the graph is a tree First we use the classi cation theorem of nite simple groups and weshow that A5 is the only nonabelian simple group whose character graph contains no triangles Then we prove that a group must be isomorphism to A5 if it is perfect and its character graphhas no triangles In addition we argue that a group must be perfect if it is nonsolvable andits character graph dose not have a triangle Finally we prove a nite nonsolvable group whosecharacter graph has no triangles is isomorphism to A5

بيژن طائري

محمد مشكوري

عليرضا عبدالهي، عاطفه قرباني