Research On The Dimensional Power Graphs Of Scattered Simple Groups And Their Irreducible Eigenvalues

As we all know,the properties of characters of a finite group G are closely related to the structure of G.For this reason,the character theory of finite groups has been widely studied.In this paper,we focus on the relationship between the structure of G and the set cd(G)of irreducible complex character degrees of G.We know that many interesting results have been given in this respect.For example,Ito-Michler's theorem shows that if every irreducible character degree of G is not divisible by a same prime p where p | |G|,then G has a normal abelian Sylow p-subgroup;Thompson's theorem shows that if every nonlinear irreducible character degree of G is divisible by a same prime p,then G has a normal p-complement.Generally,a finite group G can not be determined uniquely by the set cd(G)of irreducible character degrees of G.For instance,there are only two nonabelian groups of order 8 up to isomorphism i.e.,D8 and Q8,but cd(D8)=cd(Q8)={1,2}.Relative to the general finite groups,B.Huppert proposed a meaningful conjecture on finite nonabelian simple groups in 2000:If G has the same set of irreducible character degrees as a finite nonabelian simple group S,i.e.,cd(G)=cd(S),then G?S × A where A is an abelian group.J.P.Zhang,B.Huppert,T.P.Wakefield,H.P.Tong-Viet et al.have proved that Huppert's conjecture is right except some simple groups of Lie type and some alternating groups.In the meantime,G.Y.Chen,C.G.Shao,H.J.Xu,Y.X.Yan,B.Khosravi,N.Ahanjideh,S.Heydari et al.add the condition“|G|=|S|”and weaken the hypothesis "cd(G)=cd(S)".They consider that whether or not every finite nonabelian simple group can be determined uniquely by its order and less information of their character degrees.In this paper,we further study this problem.For this purpose we define a new undirected graph about the set of irreducible character degrees,i.e.,the degree prime-power graph.For every positive integer n and prime p,let the p-part of n be np:=max{p?|p?|n,??Z}.Let ?(G)the set of primes dividing some degrees in cd(G)and pep(G):=max{?(1)p|??Irr(G)}.The degree prime-power graph T(G)of G is defined as follows:the vertex set is V(G):={pep(G)| p ? ?(G)},and there is an edge between vertices a and b if and only if a·b divides some irreducible character degree of G.Obviously,The degree prime-power graph T(G)gives only a partial knowledge of cd(G).In this paper,we consider that whether or not every finite nonabelian simple group can be determined uniquely by its order and degree prime-power graph,.i.e.,for every finite nonabelian simple group S,if |G|=|S| and ?(G)=?(S),then whether or not G?S.For this problem,we mainly consider the all sporadic simple groups in this paper.Firstly,in Chapter 3,we study the relations between the vertex set V(G)of ?(G)and the structures of G:if the degree prime-power graph of G has a same vertex set as the degree prime-power graph of some sporadic simple group S,i.e.,V(G)=V(S),then G is nonsolvable.Then we define the nonsolvable-structure-factor vector of a nonsolvable group.In Chapter 4,we prove that the all sporadic simple groups can be determined uniquely by its order and degree prime-power graph.In order to explore whether or not more simple groups can be determined uniquely by its orders and degree prime-power graphs,we study some simple groups except all sporadic simple groups in Chapter 5.We prove that all K3-simple groups and two kinds of two-dimensional linear groups:L2(p)(where p?5)and L2(p-1)(where p?5,p=22n+1,n ?N*)can be determined uniquely by its orders and degree prime-power graphs.