On Minimal Non S-Groups

Let G be a finite group,H ≤ G.H is called a self-normal of subgroup G,if H = NG(H);H is called an S-quasinormal subgroup of G,if H permutes every Sylow subgroup of G;G is called an S-group if all subgroups of G are S-quasinormal or self-normalizing.G is called a minimal non S-group if G is not an S-group,but all proper subgroups of G are S-groups.In this paper,we main discuss the structure of minimal non S-group and give a sufficient condition for the solvability of finite groups.The thesis is divided into two section according to contents.In the first section,we discuss the structure of minimal non S-group.In the second section,we give a sufficient condition for the solvability of finite groups.We obtain some main results as follows:Theorem 2.1.1 Let G be a minimal non S-group.Then G is solvable.Theorem 2.1.2 Let G be a minimal non S-group.Then |π(G)| ≤ 3.Theorem 2.1.3 Let G be a minimal non S-group with |π(G)| = 2.Then one of the following holds.(1)G = Cq ×(Cpn × Cp),Φ(Cpn)Cp = Z(G).(2)G = Cq × Q8,Q8 induces an automorphism of order 2 on Cq.(3)G = Cqn × Cpm,m ≥ 2,Φ(Φ(P))= Z(G).(4)G =<a,b,c | aq = bq = cpm =1,ab=ba,ac = ai,bc = bj,i(?)j(mod q),ip≡jp ≡ 1(mod q)>.(5)G =<a,b,c | aqm = bqm = 1,cpn = 1,ab = ba,ac = an,bc = bn.u(?)v(mod qm),u ≡ v(mod qm-1),up ≡ vp ≡ 1(mod qm),u(?)1(mod q),v(?)1(mod q),m ≥ 2>.(6)G = Q × P,Q is an elementary abelian q-group.P is cyclic,P acts irreducibly on Q,Φ(p)induces a power automorphism of order p on Q,and Φ(Φ(P))= Z(G).(7)G = P × Cqm,Φ(Cqm)= Z(G).P is an elementary abelian p-group(8)G = Q8 × C3mTheorem 2.2.4 Let G be a finite group.If the index of its non-nilpotent maximal subgroups are square of primes or primes.Then G is a solvable group.