子群的某些性质对有限群结构的影响
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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
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.
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
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.
引文
[1] 何鸣,张雪梅,缪龙.子群的π-可补性对群结构的影响[J].扬州大学学报,2005年2月,Vol.8,No.1,1-3.
[2] GUO W B.SHUM K P,SKIBA A N,Criterions of supersolvability for products of supersolvable groups[J]. Siberian Math,2004,45 (1): 128-133.
[3] 徐明曜.有限群导引(上,下册)[M].北京:科学出版设,2001年.
[4] O.Ore,Contributions in the theory of groups of finite order [J].Duke Math. J.5,1939,431-430.
[5] 王品超,温凤桐,杨明升.幂零群的若干充分条件[J].数学进展.1998年8月,Vol.27,No.4,331-334.
[6] 陈重穆.内-外∑与极小非∑群[M].重庆:西南师范大学出版社,1988年.
[7] W.Guo.Theory of Classes of Groups[M].Science Press-Kluwer Academeic Publishers,Beijing-New York-Dordeecht-Boston-London,2000.
[8] N.Ito and J.Szep,Uber die Quasionmalteiler von endlichen Gruppen[J]. Act.Sci.Math,23,1962,168-170.
[9] W.Guo.The influence of minimal subgroups on the structure of finite groups [J].Southeast Asian Bulletin of Mathematics,22(1998),287-290.
[10] 王品超,关于有限群的两个超可解子群之积的问题[J].数学研究与评论,1993年,Vol.13,No.2,279-282.
[11] 张勤海,赵俊英,超可解群的若干充分条件[J].数学杂志,2005年,Vol.25,No.1,399-404.
[12] 缪龙,於遒,朱路进,极小子群对有限群构造的影响[J].扬州大学学报,2003年,Vol.6,No.4,1-3.
[13] 赵艳萍,S-半正规子群与超可解子群[J].曲阜师范大学学报,1993年,Vol.19,No.3,19-22.
[14] 张远达等.幂零与可解之间[M].武汉:武汉大学重版社,1988年.
[15] 路在平,王品超,某些子群是半正规的有限群[J].数学学报,1995年, Vol.41,No.5,949-954.
[16] 王品超,超可解群的若干充分条件[J].数学学报,1990年,Vol.33,No.4,480-485.
[17] WANG Yan-ming,c-normality of groups and its properties[J].Journal of Algebra, 1996,78:954-965.
[18] ROBINSON D.J.S,A course in the theory of groups[M].NY:Springer-Verlag,1982.
[19] Kurzweil,H.,Endliche gruppen[M].Spring-Verlag,Berlin,1977.
[20] 张勤海,子群为次正规或拟正规的有限群[J].山西师大学报(自),4(1991),9-11.
[21] W.E.Deskins,On maximal subgroups[J].Proc.symp.in pure Math,1959,1,100-104.
[22] 王品超,杨兆兴,有限群的某些定理[J].数学进展,1995年12月,Vol.24,No.6,547-549.
[23] B.Huppert.Endliche Gruppen I[M].Springer-Verlag(1983).
[24] 王品超,杨兆兴,有限群的极大子群的正规指数[J].工程数学学报,1994年,Vol.11,No.1,42-48.
[25] 郭秀云,有限群为超可解的充要条件[J].数学杂志,1989年,Vol.9,No.2.
[26] 苏向盈,有限群的半正规子群[J].数学杂志,8:1(1988),5-9.
[27] O.H.Kegel, On qusionormal subgroups of finite groups [J].Proc. Internet. Conf.Theorem of Groups.(Canberra,1965),209-215,Gordor and Breach,New York,1967.
[28] 朱路进,缪龙,张新建,有限群的弱c-正规[J].扬州大学学报,2003,Vol.5,No.3.8-10.
[29] W.E.Deskins,On qusionormal subgroups of finite groups[J].Math.2,82,1963,125-132.
[30] DOERK,HAWKES TO,Finite soluble groups[M].Berlin/New York:Walter de Gonvter,1992,47.
[2] GUO W B.SHUM K P,SKIBA A N,Criterions of supersolvability for products of supersolvable groups[J]. Siberian Math,2004,45 (1): 128-133.
[3] 徐明曜.有限群导引(上,下册)[M].北京:科学出版设,2001年.
[4] O.Ore,Contributions in the theory of groups of finite order [J].Duke Math. J.5,1939,431-430.
[5] 王品超,温凤桐,杨明升.幂零群的若干充分条件[J].数学进展.1998年8月,Vol.27,No.4,331-334.
[6] 陈重穆.内-外∑与极小非∑群[M].重庆:西南师范大学出版社,1988年.
[7] W.Guo.Theory of Classes of Groups[M].Science Press-Kluwer Academeic Publishers,Beijing-New York-Dordeecht-Boston-London,2000.
[8] N.Ito and J.Szep,Uber die Quasionmalteiler von endlichen Gruppen[J]. Act.Sci.Math,23,1962,168-170.
[9] W.Guo.The influence of minimal subgroups on the structure of finite groups [J].Southeast Asian Bulletin of Mathematics,22(1998),287-290.
[10] 王品超,关于有限群的两个超可解子群之积的问题[J].数学研究与评论,1993年,Vol.13,No.2,279-282.
[11] 张勤海,赵俊英,超可解群的若干充分条件[J].数学杂志,2005年,Vol.25,No.1,399-404.
[12] 缪龙,於遒,朱路进,极小子群对有限群构造的影响[J].扬州大学学报,2003年,Vol.6,No.4,1-3.
[13] 赵艳萍,S-半正规子群与超可解子群[J].曲阜师范大学学报,1993年,Vol.19,No.3,19-22.
[14] 张远达等.幂零与可解之间[M].武汉:武汉大学重版社,1988年.
[15] 路在平,王品超,某些子群是半正规的有限群[J].数学学报,1995年, Vol.41,No.5,949-954.
[16] 王品超,超可解群的若干充分条件[J].数学学报,1990年,Vol.33,No.4,480-485.
[17] WANG Yan-ming,c-normality of groups and its properties[J].Journal of Algebra, 1996,78:954-965.
[18] ROBINSON D.J.S,A course in the theory of groups[M].NY:Springer-Verlag,1982.
[19] Kurzweil,H.,Endliche gruppen[M].Spring-Verlag,Berlin,1977.
[20] 张勤海,子群为次正规或拟正规的有限群[J].山西师大学报(自),4(1991),9-11.
[21] W.E.Deskins,On maximal subgroups[J].Proc.symp.in pure Math,1959,1,100-104.
[22] 王品超,杨兆兴,有限群的某些定理[J].数学进展,1995年12月,Vol.24,No.6,547-549.
[23] B.Huppert.Endliche Gruppen I[M].Springer-Verlag(1983).
[24] 王品超,杨兆兴,有限群的极大子群的正规指数[J].工程数学学报,1994年,Vol.11,No.1,42-48.
[25] 郭秀云,有限群为超可解的充要条件[J].数学杂志,1989年,Vol.9,No.2.
[26] 苏向盈,有限群的半正规子群[J].数学杂志,8:1(1988),5-9.
[27] O.H.Kegel, On qusionormal subgroups of finite groups [J].Proc. Internet. Conf.Theorem of Groups.(Canberra,1965),209-215,Gordor and Breach,New York,1967.
[28] 朱路进,缪龙,张新建,有限群的弱c-正规[J].扬州大学学报,2003,Vol.5,No.3.8-10.
[29] W.E.Deskins,On qusionormal subgroups of finite groups[J].Math.2,82,1963,125-132.
[30] DOERK,HAWKES TO,Finite soluble groups[M].Berlin/New York:Walter de Gonvter,1992,47.