G’<G的内-FC群的几个等价条件

SOME EQUIVALENT CONDITIONS OF THE MINIMAL NON-FCGROUPS WITH THE PREFER COMMUTATOR SUBGROUPS

给出了内-IN群和内-IA群的基本分类,并得到了G’<G的内-FC群的判定定理:定理 设G为G’<G的群.那么下列条件等价:(1)G为内-FC群.(2)G为内-FA群.(3)G为具有极大子群的内-FN群.(4)G为具有指数有限的真子群的内-FN群.(5)G为具有极大子群且中心有限的内-IN群.(6)G为中心有限的内-IA群.

In this paper wefirst giveout the basic chassification of inner-IN groups and inner-1A groups and then prove the following theorem for the criterion of the inner-FC group G with G,<G.Theorem Let G be a group with C'<C. Then the following conditions are equivalent;(1) G is an inner-FC group.(2) G is an inner-FA group.(3) G is an inner-FN group which has a maximal subgroup.(4) G is an inner-FN group which has a proper subgroup with finite index.(5) G is an inner-IN group with finite center and a maximal subgroup.(6) G is an inner-IA group with finite center.