Finite Groups With Exactly Two Nonlinear Monolithic Characters

A group G is said to be a monolith if it contains only one minimal normal subgroup. A characterχof G is said to be monolithic ifχ∈IrrG and G/Kerχis a monolith. Set Trr_m(G) = {χ∈Trr(G) |χis monolithic}.Irr_m(G) influences mostly the structure of a finite group by the studies of I.Isaacs, Y.Berkovich and others. And in this paper we prove consider the groups which have exactly two nonlinear monolith characters .In this paper, we proved the following theorems:Theorem 3.1 Let a finite group G with exactly two nonlinear monolith characters, then G is sovable.Theorem 3.2 A finite group G have exactly two nonlinear monolith charactersif and only if one of the following occurs(1) G = P×A with abelian complement H, P is an extraspecial 3-group.(2) G = P×A, P is a 2-group, and P has a normal series (?) {1}, |P'| = 2, |Z(P)| = 4, where P/Z(P) is elementary abelian , A is a abelian 2-complement.(3) G = P×A, P is a 2-group, and P has a normal series (?) {1}, where P/Z{P) is an extraspecial 2-group , and A is a abelian 2-complement, |P'| = 4, |Z(P)| = 2.(4) G/Z(G) = (C_(p^m-1)/2, E(p^m)), G'∩Z(G) = {1}, G' is a minimal normal group of G, and G＝G'L×P, P∈Syl_p(Z(G)), L is a abelian p-complement. (5) G = (Q₈×Zm) (?) Q, with a Hall q'-subgroup Q₈×Z_m , which acts fix-point-freely on N = Z(Q) = Q' =Φ(Q). And Q₈×Z_m acts transitively on N^#.λis fully ramified with respect to Q/N for every A 6 (Irr(N))^#.(6) G/K is the (5), and K≤Z(G), K∩G' = {1}.(7) G has a normal series (?), where G/Z₁ = (C_p^a-1, E(p^a)), Z₁/N = Z(G/N) and N is an elementary abclian q-group. |G| = q^b(q^c -1), c≤b. The actionof the Hall q'-subgroup of G on N is Frobenius and transitively on N^#.λis fully ramified with respect to Q/N for every A€(/rr(Ar))*. where Q∈Syl_q(G). If p = q, then Q(?)G,N = Z(Q).(8) G/K is the (7), and Z(G)≠{1}, K≤Z{G), K∩G'= {1}.(9) G = HK, H∩K = Z₁≤Z{G), Z(G)∩G' = {1}, Z₁≤H, K (?) G, and H/Z₁≌ES(n,2), K/Z₁≌(C_p^a-1 E(q^a)).