Character Degrees And The Structures Of Solvable Groups

It is an important problem that research the structure from character degree graphs. After 1985 there were many outcomes, See [5], [9], [14], [15], [16], [17], [18], [24]. More recently Mark L.Lewis in [10] gave a complete classification of the following special case: the group G is solvable and P(G) has 2 connected components. In [11] Mark L. Lewis prove the Fitting height is at most 4 of G which has the condition (*).(*): disjoint union where , Assume that no prime in is adjacent in (G) to any prime in In this paper we improved it based on [11] to consider the solvable groups with the condition (*). we get the following the theorem:Theorem 3.1 Let G be a finite solvable group, and it satisfaction the condition(*).then 2 < n(G) < 4, dlp >(G) < 6. And G is one of example 1-20 . ( see the third section)In 1998 I.M.Isaacs and G.Knutson studied the effect of the set Irr{G\N) = {x Irr(G)\N do not be included in ker\} to the normal subgroup N in [7].When N G, We have Irr(G) = Irr{G/N) Irr{G|N). But the character set Irr{G/N) can only decide the structure of G/N, So the structure of the normal subgrup N and the expansion of TV to G should be decided by Irr(G\N). Now we have many results been gotten about Irr(G|N), Such as [18], [7], [8] etc. . We consider the effect of IBrP(G|N) to the structure of the normal subgroup N and the extention of N to G. We get the following :Theorem 4.1.5 let N G , G/N is a p -group. Then the prime to arbitrary non-linear have the normal Sylow p-subgroup.Theorem 4.3.2 Let G is a p-solvable group, G/N is a {(p', q'}-group, N G. the prime to all non-linear Then N have a normal q-complement.Theorem 4.3.4 Let G is p-solvable group, G/N is a p'-group, the subgroup N G. the prime to all non-linear . Then N have a normal g-complement.In 1957 B.Huppert [6] was the first to give a logarithmic bound for dl(G).J.Dixon [2] in 1986 and O.Manz,T.R.Wolf [18] in 1993 had improved it, We will "improve" it in this paper exclude some case:Theorem 5.1 Let G be solvable .(a)If G < Sn, Then dl(G) < (7/3) log3(n).unless ( primitive permutation group on the finite set f2,|Q| = 9), where dl(G) = 5.(b)Let V 0 be a faithful and completely reducible module over an arbitrary field F. Set n = dimF(V), Then dl(G) <8 + (7/3)log3(n/8).