Some Results On Linear Limits Of Characters

In this article,the theory of linear limits of character triples introduced by Dade and Loukaki in 2004 was studied.The concept of Fitting subtriples was introduced and a structure theorem of character triples was obtained.In particular,it was proved that if a triple has no nilpotent linear limit,then a linear limit of this triple contains an anisotropic controlled character-five(in the sense of Isaacs).As an application,the well-known theorem of monomial characters due to Dade was strengthened.The main conclusions of this paper are as follows:Theorem A Suppose that(?)=(G,N,?)is a linearly irreducible triple and(Z,? is the central character pair of(?).If(?)is faithful,G is solvable,and N is nonabelian,then the following hold.(1)Z = Z(F(N))<F(N),and F(N)/Z is abelian.(2)(is fully ramified with respect to F(N).If ? ? Irr(F(N)?),then the triple(?)*=(G,F(N),7)is also linearly irreducible.In this case,(?*)is a Fitting subtriple of(?)and l =(G,F(N),Z,?,?)is an anisotropic Isaacs' character-five.(3)For every prime divisor p of |F(N):Z|,Op(N)is a generalized extraspecial p-group.In particular,every Sylow subgroup of F(N)/Z is an elementary abelian group.(4)If the triple(?)is not nilpotent,i.e.N/Z is not nilpotent,then there exists an anisotropic character-five le =(G,E,Z(E),?,?)where E(?)G is an extraspecial p-subgroup,for some prime p,? ? lr(Z(EE)lies under?,and ? ?Irr(E|?)lies under ?.Furthermore,there exists a p',-subgroup S ? F(N),such that ES(?)G and CE/z(E)(S)= 1,i.e.le is a controlled character-five.We also have G = ENG(S)and E ? NG(S)= CE(S)= E ? Z = Z(E).Theorem B Let(?)(G,N,?)be a triple,where G is solvable,and assume that there exists a monomial character X ? Irr(G|?),such that x(1)be a power of an odd prime p.Then(?)has a nilpotent linear limit.Corollary C Let G be a solvable group and let X ? Irr(G)G e a monomial character,such that X(1)is a power of an odd prime p.If N<G,then any ? ? Irr(N)lying under X is a monomial character.