Finite Groups With A Special Irreducible Nonlinear Character

Let G be a finite group with a unique irreducible character ? such that ?(1)2 ||G:ker?|.This paper proved that G is a solvable group and illustrated the structure of G:1.When ker?=1,it shown that G=P × L is a minimal nonnilpotent group.Furthermore,in the case where L is abelian,it proved that G is a Frobenius group of order pn(pn-1),and in the case of cl(L)?2,?(1)=|L|.2.When ker? ?1,it shown that ker? is nilpotent and G=P×L,where P is a Sylow p-subgroup and L is a nilpotent Hall p'-subgroup.Furthermore,suppose L?kerx ?1,then L is Sylow q-subgroup and |L/L?ker?| q.In the end,it proved that cl(P)?2,what's more,if P is abelian,P ? ker?=1;if P is non-abelian,P is a special p-group and Z(P)=P'?ker? ? Z(G).The main results areTheorem 3.1 Let G be a finite group.If G has a unique irreducible character? such that ?(1)2 | |G:ker?|,then G is solvable.Theorem 4.1 Let G be a finite group and G has a unique irreducible character? such that ?(1)2 | |G:ker?1,then G=P×L,where P is a Sylow p-subgroup and L is a nilpotent Hall p'-subgroup.And the following results hold:(1)If L ? ker? ?1,then L is Sylow q-subgroup and |L/L ? ker?|= q.(2)If ker?x?1,then ker? and Pker? are nilpotent.(3)cl(P)?2.If P is abelian,then P ? ker?=1;if P is non-abelian,P is a special p-group and Z(P)=P'?ker?? Z(G).Theorem 4.2 Let G be a finite group,G has a unique irreducible character? such that ?(1)2 | |G:ker?| and ker?=1,then G=P×L,where P is a Sylow p-subgroup and L is a nilpotent Hall p'-subgroup.If cl(L)?2,then ?(1)=|L|.