On Core-finite And Its Dual Problems

Let G be a group, H a subgroup of G. We say that G is core-finite if each subgroup H of Gsatisfies that H/HG is finite. Dually, in [3], one says that G is an S*(A*', C*)-group, if each(abelian, or cyclic)subgroup H of G satisfies that |HG : H| < ∞; Further, one says that G is anS* (n)(A* (n), C*(n))-group, if each (abelian, or cyclic)subgroup H of G satisfies that|HG:H| .If one of a and b is an involution, then G is not a simple group.Theorem 2.3 The inner-finite infinite simple groups are all inner-soluble groups.Theorem 2.4 Let G be a non-abelian inner-finite group, each non-trivial proper subgroup of G is prime order cyclic group if and only if G is a simple group; each proper subgroup of G is nilpotent; and each non-trivial subgroup of G is self-normalizer.Theorem 2.5 Let G be an infinite simple group that satisfies maximal condition. G is an inner-finite group and each non-trivial proper subgroup of G is abelian if and only if for each x in G, CG (x) is the only maximal subgroup that contain x.S*(A*, C*)-groups can be regarded as a generalizations of Dedekind groups, since all of Dedekind groups are S*(A*, C*)-groups. Because the nilpotent class of Dedekind groups is at best 2, we mainly study nilpotent structures of S(A*, C*)-groups in section 3. Since the locally nilpotent S*(p)-group is nilpotent(see property 3.2),we maily study the locally nilpotent C*(p)-group in the following.Depending on [1], for finite p-group we dually obtain the nilpotent class of S*(A*, C*)-groups and the structure of their derived subgroup. So we can easily obtain nilpotent class of S*(A*, C*)-groups and the structure of their derived subgroup in the case oftorsion and locally nilpotent group. Mainly results are in the following.Theorem 3.1 If G is C*(p)-group and the exponent of G is p,then the nilpotent class of G is at most 2 and G = p.Theorem 3.2 If G is C*(p)-group,then:(l) the nilpotent class of G is at most 2;(2) the derived subgroup of G is abelian.Theorem 3.3 If G is C (p)-group and p is an odd prime,then the derived subgroup of G is elementary abelian p-group.Theorem 3.4 If G is C*(p)-group and p>3,then the nilpotent class of G is at most 2.Theorem 3.6 Let G be C*(p)-group.If p>2 and the nilpotent class of G is 3,then/?=3 and the exponent of G is 3.Theorem 3.9 If G is locally finite p-group and C*(p)-group,then the nilpotent class of G is at most 3 and the derived subgroup of G is elementary abelian p-group.Theorem 3.10 If G is locally nilpotent group and C*-group,then G is Baer nilpotent and hypercentral group.Theorem 3.11 If G is torsion locally nilpotent group and C (p)-group,then the nilpotent class of G is at most 3.In section 4, in the case of finite and infinite non-nilpotent we mainly study the soluble structure of S (A , C )-groups. Mainly results are in the following.Theorem 4.1 Finite C (p)-group G is soluble.Theorem 4.2 Finite C*(pq)-group G is soluble.Theorem 4.4 If G is finitely generated and C*(n)-group,then G is polycyclic group.Theorem 4.5 If locally soluble group G is C*(w)-group,then the chief factor of G is elementary abelian.