射影线性群区传递作用于5-（q+1，7，λ）设计
摘要
群在抽象代数中具有基本的重要地位,许多代数结构,包括环、域和模等可以看作是在群的基础上添加新的运算和公理而形成的。群的概念在数学的许多分支都有出现,而且群论的研究方法也对抽象代数的其它分支有重要影响。
本文旨在讨论特殊射影线性群PSL(2,q)和一般射影线性群PGL(2,q)区传递作用下的5-(q+1,7,λ)设计的存在性问题。本文共分为三个部分：
第一部分,我们对群与组合设计的历史背景和研究现状进行了综述,并介绍了本文所做的主要研究内容.
第二部分,我们给出了群论与组合设计的一些基本概念,为后面章节的讨论和文章的构架打下基础.
第三部分,主要对特殊射影线性群PSL(2,q)和一般射影线性群PGL(2,q)区传递作用下5-(q+1,7,λ)设计的存在性问题进行讨论,并在设计存在的情况下,进行设计的构造。我们有以下主要结论：
定理1：设D=(X,B“)是一个5-(q+1,7,λ)设计,PSL(2,q)区传递作用在X上,X=GF(q)∪{∞},则当q=23时,存在两个不同构的5-(q+1,7,λ)设计,其中λ=3.
定理2：设D=(X,BG)是一个5-(q+1,7,λ)设计,PGL(2,q)区传递作用在X上,X=GF(q)∪{∞},则下列情形发生：
(1)当q=1 7时,存在唯一的5-(q+1,7,λ)设计,其中λ=3；
(2)当q=23时,存在唯一的5-(q+1,7,λ)设计,其中λ=6.
Groups take a basic importance in abstract algebra. Many algebraic structures, including rings, fields, and molds that can be seen as the basis of the group to add new operations and axioms formed. The concept of group of has emerged in many branches of mathematics, and the group theory methods are also has an important effect in other branches of abstract algebra.
This paper aims at discussing the existence of the 5-(q+1,7,λ) designs admitting the block transitive automorphism groups projective special linear group PSL(2,q) and projective general linear group PGL(2,q). This thesis consists of three departments.
In chapter 1, we give some introduction about the history and current research situation of the group theory and design, and we describe the major research by this article.
In chapter 2, we introduce the elementary concepts of the group theory and design that will be used in this thesis.
In chapter 3, we focus on discussing the existence of the 5-(q+1,7,λ) design admitting the block transitive automorphism groups PSL(2,q) and PGL(2,q).Then we get some designs with the existence of the 5-(q+1,7,λ) design. We have the main theorem as follows:
Theorem 1:Let D=(X,BG) is a 5—(q+1,7,λ)design admitting the block transitiVe automorphism groups PSL(2,q),X=GF(q)∪{∞}.Thern q=23, there is two 5—(q+1,7,λ)designs with not automorphism, whereλ=3.
Theorem 2:Let D=(X,BG) is a 5—(q+1,7,λ)design admitting the block transitive automorphism groups PGL(2,q),X=GF(q)u{∞).Then the following may happen: (1)q=17,D=(X,BG)is a 5—(18,7,3)design；(2)q=23,D=(X,BG)is a 5—(24,7,6)design.
本文旨在讨论特殊射影线性群PSL(2,q)和一般射影线性群PGL(2,q)区传递作用下的5-(q+1,7,λ)设计的存在性问题。本文共分为三个部分：
第一部分,我们对群与组合设计的历史背景和研究现状进行了综述,并介绍了本文所做的主要研究内容.
第二部分,我们给出了群论与组合设计的一些基本概念,为后面章节的讨论和文章的构架打下基础.
第三部分,主要对特殊射影线性群PSL(2,q)和一般射影线性群PGL(2,q)区传递作用下5-(q+1,7,λ)设计的存在性问题进行讨论,并在设计存在的情况下,进行设计的构造。我们有以下主要结论：
定理1：设D=(X,B“)是一个5-(q+1,7,λ)设计,PSL(2,q)区传递作用在X上,X=GF(q)∪{∞},则当q=23时,存在两个不同构的5-(q+1,7,λ)设计,其中λ=3.
定理2：设D=(X,BG)是一个5-(q+1,7,λ)设计,PGL(2,q)区传递作用在X上,X=GF(q)∪{∞},则下列情形发生：
(1)当q=1 7时,存在唯一的5-(q+1,7,λ)设计,其中λ=3；
(2)当q=23时,存在唯一的5-(q+1,7,λ)设计,其中λ=6.
Groups take a basic importance in abstract algebra. Many algebraic structures, including rings, fields, and molds that can be seen as the basis of the group to add new operations and axioms formed. The concept of group of has emerged in many branches of mathematics, and the group theory methods are also has an important effect in other branches of abstract algebra.
This paper aims at discussing the existence of the 5-(q+1,7,λ) designs admitting the block transitive automorphism groups projective special linear group PSL(2,q) and projective general linear group PGL(2,q). This thesis consists of three departments.
In chapter 1, we give some introduction about the history and current research situation of the group theory and design, and we describe the major research by this article.
In chapter 2, we introduce the elementary concepts of the group theory and design that will be used in this thesis.
In chapter 3, we focus on discussing the existence of the 5-(q+1,7,λ) design admitting the block transitive automorphism groups PSL(2,q) and PGL(2,q).Then we get some designs with the existence of the 5-(q+1,7,λ) design. We have the main theorem as follows:
Theorem 1:Let D=(X,BG) is a 5—(q+1,7,λ)design admitting the block transitiVe automorphism groups PSL(2,q),X=GF(q)∪{∞}.Thern q=23, there is two 5—(q+1,7,λ)designs with not automorphism, whereλ=3.
Theorem 2:Let D=(X,BG) is a 5—(q+1,7,λ)design admitting the block transitive automorphism groups PGL(2,q),X=GF(q)u{∞).Then the following may happen: (1)q=17,D=(X,BG)is a 5—(18,7,3)design；(2)q=23,D=(X,BG)is a 5—(24,7,6)design.
引文
[1]徐明耀.有限群导引[M].北京：科学出版社,2001
[2]沈灏.组合设计理论[M].上海：上海交通大学出版社,1996：1-33
[3]Michael Huber. On the Cameron-Praeger conjecture[J]. Journal of Combinatorial Theory,Series A 117(2010):196-203
[4]P.J.Cameron,G.R.Omidi,B.Tayfeh-Rezaie.3-Designs from PGL(2,q)[J].May 19,2006 Mathematics Subject Classifications:05B05,20B20
[5]P.J.Cameron,H.R.Maimani,G.R.Omidi,B.Tayfeh-Rezaie.3-designs from PSL(2,q) [J]. Elsevier Science,28 May 2004
[6]Z.Ealsmi and GB.Khosrovshahi. Some New 6-(14,7,4) Designs[J]. Journal of Combinatorial Theory,Series A 2001 (93),141-152
[7]G.B.Khosrovshahi,M.Mohammad-Noori,B-Tayfeh-Rezaie. Classification of 6-(14,7,4) Designs with Nontrivial Automorphism Groups[J].2002 Wiley Periodicals,Inc.J Combin Designs 10:180-194,2002
[8]E.S.Kramer,S.S.Magliveras,E.A.O'Brien. Some new large sets of t-designs[J]. Australasian Journal of Combinatorics 7(1993):189-193
[9]R.Laue,G.R.Omidi,B.Tayfeh-Rezaie,A.Wassermann. New Large Sets of t-Designs with Prescribed Groups of Automorphisms[J]. Published online 20 November 2006 in Wiley InterScience(www.interscience.wiley.com).DOI 10.1002/jcd.20128
[10]Y.M.Chee,C.J.Colbourn,S.C.Furino,D.L.Kreher. Large sets of disjoint t-designs[J]. The Australasian Journal of Combinatorics 2(1990):111-119
[11]L.Teirlinck. On large sets of disjoint quadruple systems[J]. Ars Combinatoria 17 (1984):173-176
[12]Guangguo Han. Unsolvable block transitive automorphism groups of 2-(v,k,l) (k=6,7,8,9) designs[J]. Discrete Mathematics 308(2008) 5632-5644
[13]A.Delandtsheer,J.Doyen,J.Siemons.C.Tamburini. Doubly homogeneous 2-(v,k,l) designs[J]. J.Combin.Theory Ser.A 43(1986),140-145
[14]M.W.Liebeck,Chery E.Praeger and J.Saxl. A classification of the maximal subgroups of the finite alternating and symmetric groups[J]. J.Algrbra 111 (1987),365-383
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[16]H.维兰特.有限置换群(王萼芳,译)[M].北京：科学出版社,1984
[17]Ian Anderson Liro Honkala. A short Course in Combinatorial Designs[M]. Internet Edition, Spring 1997
[18]M.S.Keranen,D.L.Kreher,P.J-S.Shiue. Quadruple Systems of the Projective Special Linear Group PSL(2,q), q=1(mod4) [J]. Inc.J Combin Designs 2003,11:339-351
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[20]王华国.射影线性群作用下的区传递4-设计：[硕士学位论文].长沙：中南大学,2009
[21]C.A.Cusack,S.W.Graham,D.L.Kreher. Large sets of 3-designs from PSL(2,q), with block sizes 4 and 5[J]. J combin Des 1995,3:147-160
[22]Ricard Marti,Enric Nart. Orbits of rational n-sets of projective spaces under the action of the linear group[J]. Journal of Combinatorial Theory,Series A 2008(115):547-568
[23]Niranjan Balachandram,Dijen Ray-Chaudhuri. Simple 3-designs and PSL(2,q) with q= 1(mod4) [J]. Des.Codes Cryptogr.2007(44):263-274
[24]Byeong-Kweon Oh,Jangheon Oh,Hoseog Yu. New infinite families of 3-designs from algebraic curves over F[J]. European Journal of Combinatorics 2007 (28):1262-1269
[25]M.R.Darafsheh. Designs from the group PSL2(q),q even[J]. Des Codes Crypt 2006 (39):311-316
[26]Shiro Iwasaki. Translations of the squares in a finite field and an infinite family of 3-designs[J]. European Journal of Combinatorics 2003,24:253-266
[27]Anton Betten, Reinhard Laue,Alfred Wassermann. New t-designs and large sets of t-designs[J]. Discrete Mathematics 197/198(1999):111-121
[28]G.B.Khosrovshahi and CH.Maysoori. On the Bases for Trades[J]. LINEAR ALGEBRA AND ITS APPLICATIONS 226-228:731-748(1995)
[29]Makoto Araya,Masaaki Harada. Mutually Disjoint Steiner Systems S(5,8,24) and 5-(24,12,48) Designs[J]. Mathematics Subject Classifications Jan 5, 2010:05B05
[30]Gh.R.Omidi,M.R.Pournaki,B.Tayfeh-Rezaie.3-Designs with Block Size 6 from PSL(2,q) and Their Large Sets[J].2000 Mathematics Subject Classification:Primary 05B05,05B30;Secondary 20D06
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[32]代少军.几乎单群与组合设计：[博士学位论文].长沙：中南大学,2008
[33]N.Jacobson基础代数(第一卷,第一分册)[M].上海：高等教育出版社,1987
[34]T.Beth, D.Jungickel, H.Lenz. Design theory [M]. Cambridge University Press, 1993
[35]L.E.Dickson. Linear groups, with an introduction to the Galois field theory [M]. Dover Publications,1958
[36]Y.M.Chee,C.J.Colbourn,S.C.Furino,D.L.Kreher. Large sets of disjoint t-designs[J]. The Australasian Journal of Combinatorics 2(1990):111-119
[37]C.A.Cusack. PSL(2,q) as an automorphism group of a 3-(q+1,5,λ) Design Master Thesis[M]. Michigan Thenological University,1994
[38]D.R.Hughes. On t-designs and groups[J]. Amer.J.Math.87(1965),761-778
[39]S.Iwasaki and T.Meixner. A remark on the action of PGL(2,q)and PSL(2,q) on the projective line[J]. Hokkaido Math.J.16(1997),203-209
[40]R.Laue,S.S.Magliveras and A.Wassermann. New large sets of t-designs[J]. J.Combin.Des.9(2001),40-59
[41]M.W.Liebeck,Chery E.Praeger and J.Saxl. A classification of the maximal subgroups of the finite alternating and symmetric groups[J]. J.Algrbra 111 (1987),365-383
[42]Anton betten,Reinhard laue,Alfred Wassermann. A Steiner 5-Design on 36 Points[J]. Designs,Codes and Cryptography,17,181-186(1999)
[43]R.H.F.Denniston. Some new 5-Designs[J]. Bull.London Math.Soc,8(1976),263-267
[44]E.F.Assmus,H.F.Mattson. New 5-designs[J]. J.Comb.Theory,Vol.6(1969),122-151
[45]M.J.Grannell,T.S.Griggs,R.A.Mathon. Steiner systems S(5,6, v) with v= 72 and 84[J]. Mathematics of Computation,Vol 67,Number221,January 1998,357-359
[46]S.Ajoodani-Namini. Extending large sets of t-designs[J]. J.Combin.Theory A 76 (1996),139-144
[47]W.O.Alltop. Extending t-designs[J]. J.combin.Theory A 18(1975)177-186
[48]E.S.Kramer,D.M.Mesner. t-designs on hypergraphs[J]. Discrete Math.15(1976) 263-296
[49]B.Schmalz. The t-designs with prescribed automorphism group,new simple 6-designs[J]. J.Combin.Des.1(1993)125-170
[50]Tran van Trung. On the construction of t-designs and the existence of some new infinite families of simple 5-designs[J]. Arch.Math.47(1986)187-192
[51]Tran van Trung, Qiu-rong Wu, D.M.Mesner. High order intersection numbers of t-designs [J]. J.Statist.Plann.Inference 56(1996) 257-268
[52]Tran van Trung. On the Existence of an Infinite Family of Simple 5-Designs [J]. Math.Z.187,285-287(1984)
[53]Masaaki Harada, Masaaki Kitazume, Akihiro Munemasa. On a 5-design related to an extremal doubly even self-dual code of Iength72 [J]. Journal of Combinato-rial Theory,Series A 107(2004):143-146
[54]Michael Huber. Classification of Flag-Transitive Steiner Quadruple Systems [J]. Journal of Combinatorial Theory,Series A 94,180-190(2001)
[55]Weijun Liu, Jinglei Li. Finite projective planes admitting a projective linear group PSL(2,q) [J]. Linear Algebra and its Applications 413 (2006) 121-130
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[2]沈灏.组合设计理论[M].上海：上海交通大学出版社,1996：1-33
[3]Michael Huber. On the Cameron-Praeger conjecture[J]. Journal of Combinatorial Theory,Series A 117(2010):196-203
[4]P.J.Cameron,G.R.Omidi,B.Tayfeh-Rezaie.3-Designs from PGL(2,q)[J].May 19,2006 Mathematics Subject Classifications:05B05,20B20
[5]P.J.Cameron,H.R.Maimani,G.R.Omidi,B.Tayfeh-Rezaie.3-designs from PSL(2,q) [J]. Elsevier Science,28 May 2004
[6]Z.Ealsmi and GB.Khosrovshahi. Some New 6-(14,7,4) Designs[J]. Journal of Combinatorial Theory,Series A 2001 (93),141-152
[7]G.B.Khosrovshahi,M.Mohammad-Noori,B-Tayfeh-Rezaie. Classification of 6-(14,7,4) Designs with Nontrivial Automorphism Groups[J].2002 Wiley Periodicals,Inc.J Combin Designs 10:180-194,2002
[8]E.S.Kramer,S.S.Magliveras,E.A.O'Brien. Some new large sets of t-designs[J]. Australasian Journal of Combinatorics 7(1993):189-193
[9]R.Laue,G.R.Omidi,B.Tayfeh-Rezaie,A.Wassermann. New Large Sets of t-Designs with Prescribed Groups of Automorphisms[J]. Published online 20 November 2006 in Wiley InterScience(www.interscience.wiley.com).DOI 10.1002/jcd.20128
[10]Y.M.Chee,C.J.Colbourn,S.C.Furino,D.L.Kreher. Large sets of disjoint t-designs[J]. The Australasian Journal of Combinatorics 2(1990):111-119
[11]L.Teirlinck. On large sets of disjoint quadruple systems[J]. Ars Combinatoria 17 (1984):173-176
[12]Guangguo Han. Unsolvable block transitive automorphism groups of 2-(v,k,l) (k=6,7,8,9) designs[J]. Discrete Mathematics 308(2008) 5632-5644
[13]A.Delandtsheer,J.Doyen,J.Siemons.C.Tamburini. Doubly homogeneous 2-(v,k,l) designs[J]. J.Combin.Theory Ser.A 43(1986),140-145
[14]M.W.Liebeck,Chery E.Praeger and J.Saxl. A classification of the maximal subgroups of the finite alternating and symmetric groups[J]. J.Algrbra 111 (1987),365-383
[15]W.J.Liu. Block transitive 2-(v,k,l) designs. Ph.D. Thesis,Zhejiang University, 1998
[16]H.维兰特.有限置换群(王萼芳,译)[M].北京：科学出版社,1984
[17]Ian Anderson Liro Honkala. A short Course in Combinatorial Designs[M]. Internet Edition, Spring 1997
[18]M.S.Keranen,D.L.Kreher,P.J-S.Shiue. Quadruple Systems of the Projective Special Linear Group PSL(2,q), q=1(mod4) [J]. Inc.J Combin Designs 2003,11:339-351
[19]Van Leijenhorst C.D. Orbits on the Projective Line[J]. Journal Of Combinatorial Theory, Series A 1981,31:146-154
[20]王华国.射影线性群作用下的区传递4-设计：[硕士学位论文].长沙：中南大学,2009
[21]C.A.Cusack,S.W.Graham,D.L.Kreher. Large sets of 3-designs from PSL(2,q), with block sizes 4 and 5[J]. J combin Des 1995,3:147-160
[22]Ricard Marti,Enric Nart. Orbits of rational n-sets of projective spaces under the action of the linear group[J]. Journal of Combinatorial Theory,Series A 2008(115):547-568
[23]Niranjan Balachandram,Dijen Ray-Chaudhuri. Simple 3-designs and PSL(2,q) with q= 1(mod4) [J]. Des.Codes Cryptogr.2007(44):263-274
[24]Byeong-Kweon Oh,Jangheon Oh,Hoseog Yu. New infinite families of 3-designs from algebraic curves over F[J]. European Journal of Combinatorics 2007 (28):1262-1269
[25]M.R.Darafsheh. Designs from the group PSL2(q),q even[J]. Des Codes Crypt 2006 (39):311-316
[26]Shiro Iwasaki. Translations of the squares in a finite field and an infinite family of 3-designs[J]. European Journal of Combinatorics 2003,24:253-266
[27]Anton Betten, Reinhard Laue,Alfred Wassermann. New t-designs and large sets of t-designs[J]. Discrete Mathematics 197/198(1999):111-121
[28]G.B.Khosrovshahi and CH.Maysoori. On the Bases for Trades[J]. LINEAR ALGEBRA AND ITS APPLICATIONS 226-228:731-748(1995)
[29]Makoto Araya,Masaaki Harada. Mutually Disjoint Steiner Systems S(5,8,24) and 5-(24,12,48) Designs[J]. Mathematics Subject Classifications Jan 5, 2010:05B05
[30]Gh.R.Omidi,M.R.Pournaki,B.Tayfeh-Rezaie.3-Designs with Block Size 6 from PSL(2,q) and Their Large Sets[J].2000 Mathematics Subject Classification:Primary 05B05,05B30;Secondary 20D06
[31]廖小莲.几乎单群与斯坦诺4-设计：[硕士学位论文].长沙：中南大学,2006
[32]代少军.几乎单群与组合设计：[博士学位论文].长沙：中南大学,2008
[33]N.Jacobson基础代数(第一卷,第一分册)[M].上海：高等教育出版社,1987
[34]T.Beth, D.Jungickel, H.Lenz. Design theory [M]. Cambridge University Press, 1993
[35]L.E.Dickson. Linear groups, with an introduction to the Galois field theory [M]. Dover Publications,1958
[36]Y.M.Chee,C.J.Colbourn,S.C.Furino,D.L.Kreher. Large sets of disjoint t-designs[J]. The Australasian Journal of Combinatorics 2(1990):111-119
[37]C.A.Cusack. PSL(2,q) as an automorphism group of a 3-(q+1,5,λ) Design Master Thesis[M]. Michigan Thenological University,1994
[38]D.R.Hughes. On t-designs and groups[J]. Amer.J.Math.87(1965),761-778
[39]S.Iwasaki and T.Meixner. A remark on the action of PGL(2,q)and PSL(2,q) on the projective line[J]. Hokkaido Math.J.16(1997),203-209
[40]R.Laue,S.S.Magliveras and A.Wassermann. New large sets of t-designs[J]. J.Combin.Des.9(2001),40-59
[41]M.W.Liebeck,Chery E.Praeger and J.Saxl. A classification of the maximal subgroups of the finite alternating and symmetric groups[J]. J.Algrbra 111 (1987),365-383
[42]Anton betten,Reinhard laue,Alfred Wassermann. A Steiner 5-Design on 36 Points[J]. Designs,Codes and Cryptography,17,181-186(1999)
[43]R.H.F.Denniston. Some new 5-Designs[J]. Bull.London Math.Soc,8(1976),263-267
[44]E.F.Assmus,H.F.Mattson. New 5-designs[J]. J.Comb.Theory,Vol.6(1969),122-151
[45]M.J.Grannell,T.S.Griggs,R.A.Mathon. Steiner systems S(5,6, v) with v= 72 and 84[J]. Mathematics of Computation,Vol 67,Number221,January 1998,357-359
[46]S.Ajoodani-Namini. Extending large sets of t-designs[J]. J.Combin.Theory A 76 (1996),139-144
[47]W.O.Alltop. Extending t-designs[J]. J.combin.Theory A 18(1975)177-186
[48]E.S.Kramer,D.M.Mesner. t-designs on hypergraphs[J]. Discrete Math.15(1976) 263-296
[49]B.Schmalz. The t-designs with prescribed automorphism group,new simple 6-designs[J]. J.Combin.Des.1(1993)125-170
[50]Tran van Trung. On the construction of t-designs and the existence of some new infinite families of simple 5-designs[J]. Arch.Math.47(1986)187-192
[51]Tran van Trung, Qiu-rong Wu, D.M.Mesner. High order intersection numbers of t-designs [J]. J.Statist.Plann.Inference 56(1996) 257-268
[52]Tran van Trung. On the Existence of an Infinite Family of Simple 5-Designs [J]. Math.Z.187,285-287(1984)
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