Irr(G|N) And The Structures Of Solvable Groups

It is an important problem that research the structures from certain subsets of the characters of a group.The dissertation investigated that the set that was Irr (G| N) = { x Irr(G) | N kerx where N was a normal subgroup of G had influences on the properties and the structures of N. We proved the theorems as following:Theorem 2.2 Let N<3 G. If each character of Irr(GlN) is monomial, then N is solvable anddl (N) |ed (G|N)|.Theorem 2.6 Let N< G. If each character of Irr (G|N) has a distinct degree and G is solvable, then(1) N is one of the groups as followings:(a) N is abelian;(b) N is an extraspecial 2 - group;(c) Nisa2- transitive Frobenius group having cyclic complement;(d) N is the 2 ?transitive Frobenius group of order 72 having a quaternion complement.(2) N has a characteristic series: N = No>Ni>"- >N* = 1 such that Nj/N1 + 1 is p, -group, i =0,1, ....., k - 1.Theorem 3.1 Let G be monolith and N<] G. For a prime number p, if each a€: cdmCGl N), p | a, then N is solvable and N has a normal p - complement.Theorem 3.5 Let N < G such that for each x Irr(G| N), there exists H^G and X€r Irr (H) such that X(l)