Finite Groups With A Special Irreducible Nonlinear Character

Let G be a finite group with a unique irreducible character χ such that χ(1)2 ||G:kerχ|.This paper proved that G is a solvable group and illustrated the structure of G:1.When kerχ=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,χ(1)=|L|.2.When kerχ ≠1,it shown that kerχ 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∩kerχ| q.In the end,it proved that cl(P)≤2,what’s more,if P is abelian,P ∩ kerχ=1;if P is non-abelian,P is a special p-group and Z(P)=P’≤kerχ ∩ Z(G).The main results areTheorem 3.1 Let G be a finite group.If G has a unique irreducible characterχ such that χ(1)2 | |G:kerχ|,then G is solvable.Theorem 4.1 Let G be a finite group and G has a unique irreducible characterχ such that χ(1)2 | |G:kerχ1,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 ∩ kerχ ≠1,then L is Sylow q-subgroup and |L/L ∩ kerχ|= q.(2)If kerχx≠1,then kerχ and Pkerχ are nilpotent.(3)cl(P)≤2.If P is abelian,then P χ kerχ=1;if P is non-abelian,P is a special p-group and Z(P)=P’≤kerχ∩ Z(G).Theorem 4.2 Let G be a finite group,G has a unique irreducible characterχ such that χ(1)2 | |G:kerχ| and kerχ=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 χ(1)=|L|.