On The CC-Subgroups, Semi-Subgroups And The Structure Of Groups.

Let G be a group and let H be a subgroup of G.H is called a CC-subgroup of G if the centralizer CG(r)≤H for every l≠x∈H. Obviously,G is a CC-subgroup of G.In this paper,the CC-subgroups are proper groups.We mainly prove the following:Theorem 3.4 Let G be a finite group.If let each CC-subgroups is a maximal subgroup of G,then |G|=pqn,where n≤p-1.Especially,if n=p-1,then G=)(?)N. where is a group of order p,N=××…×, bia=bi+1,i=1,2,…,p-2,bp-1a=b1-1b2-1…bp-1-1.Next,we study the CC-subgroups in locally finite groups.Theorem 4.1 Let G b(.a locally finite group.If G has a CC-subgroup,but each proper subgroup of G has no proper CC-subgroups,then G is an extension of an elementary abelian subgroup of order≤pq-1 by a cyclic group of order q,where p,q are prime number and(p,q)=1.Theorem 4.2 Let G be a locally finite group.If G has a CC-subgroup,but each proper infinite subgroup of G has no proper CC-subgroups,then G is an extension of divisible abelian subgroup of rank p-1 by a cyclic group of order q,where p,q are prime number and(p,q)=1.In this paper,we also study the groups which contain semi-CC-subgroups.