【作者】 陶司兴；

【导师】 王品超；

【作者基本信息】 曲阜师范大学 ， 基础数学， 2006， 硕士

【摘要】 子群的性质对群的结构有着重要的影响,通过对它们的研究可以获得关于原群结构的大量信息。本文的主要工作是在[1],[2],[3]的基础上,对群的结构进行研究。全文分为三章。 在第一章中,一方面我们利用π-可补子群的性质给出了有限群为超可解群及幂零群的若干充分条件; 例如:定理3设G是群,2∈π,如果G的每个素数阶子群包含在SE(G)中,G的每个4阶循环子群在G中π-可补,则G为超可解群。 定理7设G是群,G的素数阶子群包含在Z_∞(G)中,2∈π,如果G的每个4阶循环子群在G中π-可补,则G为幂零群。 另一方面我们研究了π-可补子群对群系的影响。 例如:定理10设F是子群闭的局部群系并具有下列性质:内F-群可解,其F-上根是一个Sylow子群。设N是G的正规子群,G/N是F-群。2∈π,如果N的每个4阶循环子群在G中π-可补且N的每个极小子群包含在G的F-超中心内,那么G是一个F-群。 在第二章中,一方面我们利用子群之间的条件置换及完全条件置换的性质给出了有限群为超可解群的若干充分条件; 例如:定理4若群G的每个Sylow子群的正规化子在G中完全条件置换,则G为超可解群。 另一方面我们利用子群之间的条件置换给出了两个群的乘积为超可解群的充分条件。 例如:定理12设H,K为G的超可解子群,G=HK,G′为幂零群,又更多还原

【Abstract】 When we do research about the structure of a group,the influence of subgroups is important.We can get a lot of information about the structure of the group from the properties of its subgroups.In this paper,the main work is to study the structure of a finite group on the basis of [1],[2],[3]. This paper is composed of three chapters.In chapter 1 ,on one hand ,we give some sufficient conditions for a finite group to be supersolvable and nilipotent by using the properties of π-supplemented subgroups;For example: Theorem 3 Let G be a group, 2 ∈ π, if every subgroup of G of prime order is contained in SE(G), every cyclic subgroup of G of order 4 is π-supplemented in G, then G will be supersolvable.Theorem 7 Let G be a group, if every subgroup of G of prime order is contained in Z_∞(G),2 ∈π, every cyclic subgroup of G of order 4 is π-supplemented in G,then G will be nilpotent.On the other hand we study the influence of π-supplemented subgroups on formation.For example: Theorem 10 Let T be a subgroup-closed local formation with the following properties :an inner F-group which is solvable and its F-residual is a Sylow subgroup .2∈6 π, if every cyclic subgroup of G of order 4 is π-supplemental in G and every minimal subgroup of G is contained in the F-hypercenter of G ,then G is an F-group.In chapter 2, on one hand ,we give some sufficient conditions for a finite group to be supersolvable by using the properties of conditional permutable and completely conditional permutable between subgroups;For example: Theorem 4 If every normalizer of Sylow subgroup of G is completely conditional permutable,then G will be a supersolvable group.On the other hand ,we give some sufficient conditions for products of two subgroups to be supersolvable by using conditional pemutable between subgroups.For example: Theorem 12 Let H, Kbe supersolvable subgroup of G, G = HK, G’is a nilpotent group ,if H is conditional permutable in K, if K is conditional permutable in i/,then G will be a supersolvable group.In chapter 3, we generalize two theorems in [3],obtain some more profound results.For example: Theorem 1 LetG be a finite group, p is the smallest prime divisor of \G\,P G Sylp(G),suppose P is an Abel group,other prime divisor of \G\ is larger than pn,then G will be a p-nilpotent group.更多还原