Solvable Groups With Maximal Normal Closure Of Subgroups

A finite group G is called MC-group,if(x)(?)G and there exist a prime p such that |G:(x))G||p for every element x of G.In this paper,we classify all MC-groups which are solvable groups.The following conclusions are derived:Theorem 3.1 Let G be a finite solvable group.If G is a non-nilpotent MC-group,then G/G’ is either a cyclic group of prime power order or the direct product of two cyclic groups.Theorem 3.2 Let G be a finite solvable group.If G is a non-nilpotent MC-group,then we get the following conclusions:(1)Suppose that G/G’ is a cyclic group of prime power order,G=G’(?)(y), where o(y)=qm.(1.1)If m=1 and G’ is an Abelian group,then G’ is an elementary Abelian p-group; or G’=(a1)×(a2),where o(a1)=pn,o(a2)=p,n≥2; or G’ is a q’-group in which all subgroups are normal in G.(1.2)If m=1 and G’ is non-Abelian group,then exp(G’)=p for p≥3; or exp(G’)≤22 for p=2.(1.3)If m≥2,then G’ is a q’-group in which all subgroups are normal in G.(2)Suppose that G/G’ is the direct product of two cyclic groups,i.e.|G/G’|= pmq,then every subgroup of G’ is normal in G.(2.1)If p=q,then G=Gp×Gp’,,where Gp’,≤G’.Gp=(x1)×(x2),o(x1)= pm,o(x2)=q or Gp is isomorphic to the group which was got by Z.Janko in[1].(2.2)If m=1 and p≠q,then G=((x1)×(x2))×G’,where o(x1)=p,o(x2):q.(2.3) If m≥ 2 and p≠q, then G=(Gp × G’) ×(x2), where Gp is a cyclic group of order pm, o(x2)= q.