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Solvable Block-Transitive Automorphism Groups of 2−(v, k, 1) Designs

Huiling Li, Weijun Liu
2001 Journal of combinatorial theory. Series A  
Let G be a solvable block transitive automorphism group of a 2 − (v; 5; 1) design and suppose that G is not ag transitive.  ...  We will prove that (1) if G is point imprimitive, then v = 21, and G 6 Z21 : Z6; (2) if G is point primitive, then G 6 A L(1; v) and v = p a , where p is a prime number with p ≡ 21 (mod 40), and a an odd  ...  Let G be a solvable block transitive automorphism group of a 2 − (v; 5; 1) design.  ... 
doi:10.1006/jcta.2000.3067 fatcat:spphi2bw7ravxg5kcnmim5ijvm

Solvable block transitive automorphism groups of 2−(v,5,1) designs

Wenwen Tong, Huiling Li
2003 Discrete Mathematics  
Let G be a solvable block transitive automorphism group of a 2 − (v; 5; 1) design and suppose that G is not ag transitive.  ...  We will prove that (1) if G is point imprimitive, then v = 21, and G 6 Z21 : Z6; (2) if G is point primitive, then G 6 A L(1; v) and v = p a , where p is a prime number with p ≡ 21 (mod 40), and a an odd  ...  Let G be a solvable block transitive automorphism group of a 2 − (v; 5; 1) design.  ... 
doi:10.1016/s0012-365x(02)00763-x fatcat:ul6rtui3sfcltgqdn2olb6lufu

Unsolvable block transitive automorphism groups of 2-(v,k,1)(k=6,7,8,9) designs

Guangguo Han
2008 Discrete Mathematics  
In this paper, we study block transitive automorphism groups of 2-(v, k, 1) block designs.  ...  Let D be a 2-(v, k, 1) (k = 6, 7, 8, 9) design admitting a block transitive, point primitive but not flag transitive group G of automorphisms.  ...  Camina and Siemons [6] classified 2-(v, 4, 1) designs with a block transitive, solvable group of automorphisms.  ... 
doi:10.1016/j.disc.2007.10.051 fatcat:7u5jnqktjnhgvk2boxpva5dvsa

Block transitive 2-(v,k,1) designs and two-dimensional projective linear groups

Shaojun Dai, Jing Chen, Zhenzhen Song
2015 International Journal of Algebra  
Let G act as a block-transitive and point-primitive automorphism group of a non-trivial design D with v points and blocks of size k. Set k 2 = (k, v1).  ...  This article is a contribution to the study of the automorphism groups of 2 − (v, k, 1) designs.  ...  Liu [6] classified 2 − (v, k, 1)(where k = 6, 7, 8, 9, 10) designs with a block transitive, solvable group of automorphisms.  ... 
doi:10.12988/ija.2015.528 fatcat:4cybp4qayvgwroobfkx6vexbwu

Nonexistence of $2-(v, k, 1)$ designs admitting automorphism groups with socle $E_8(q)$

Shangzhao Li
2015 Miskolc Mathematical Notes  
In particular, the existence of 2 .v; k; 1/ designs admitting an interesting group of automorphisms is of great interest.  ...  Thirty years ago, a six-person team classified 2 .v; k; 1/ designs which have flag-transitive automorphism groups.  ...  ACKNOWLEDGEMENT The author is grateful to the editor and reviewer for pointing out errors in the original version of this paper and their valuable comments and suggestions.  ... 
doi:10.18514/mmn.2015.1473 fatcat:w4k7qonjyfdntp6cxjgqmcl6g4

The Ree groups 2G2(q) and 2−(v,k,1) block designs

Shenglin Zhou, Huiling Li, Weijun Liu
2000 Discrete Mathematics  
Let G act as a block primitive automorphism group on a 2 − (v; k; 1) design, in this paper we proved that if 2 G2(q) E G6Aut( 2 G2(q)), then G is point primitive.  ...  Let G act as a block transitive group of automorphisms of a 2 − (v; k; 1) design. Let B be a block and H a subgroup of G B .  ...  We will always assume that 26k ¡ v. Let G6Aut D be a group of automorphisms of a 2 − (v; k; 1) design D. G is said to be block transitive (block primitive) on D if G is transitive (primitive) on B.  ... 
doi:10.1016/s0012-365x(00)00050-9 fatcat:bcucw7iejfcqpbcgd4eszmteny

Automorphism groups of designs

William M. Kantor
1969 Mathematische Zeitschrift  
If F is a flag-transitive automorphism group of a design, then i) if (v-1, k-1)=1 then F is 2-transitive on points; and ii) f (v-t,k-1)=2 then F is either 2-transitive on points or has rank 3 on points  ...  Let F be a sharply flag-transitive, sharply 2-homogeneous automorphism group of a design ~ with (v, k)= 1.  ... 
doi:10.1007/bf01111409 fatcat:3xuznnemwncpjnmkssqtgpqf5a

Block-transitive, point-primitive Steiner 3-designs [article]

Yunsong Gan, Weijun Liu
2021 arXiv   pre-print
This paper studies the long-standing open problem of the reduction of Steiner 3-designs admitting a block-transitive automorphism group.  ...  We prove that if G acts as a point-primitive, block-transitive automorphism group of a nontrivial Steiner 3-design, then G is of affine or almost simple type.  ...  Huber, The classification of flag-transitive Steiner 3-designs, Adv. Geom. 5 (2005) 195–221. [8] H.L. Li, W.J. Liu, Solvable block-transitive automorphism groups of 2-(v, k, 1) designs, J.  ... 
arXiv:2112.00466v3 fatcat:r3z4iaxfwnaklbma4hzzrsal3m

The 2-(v,11,1) Designs with Unsolvable Block Transitive Automorphism

Kun Zhao, Shaojun Dai
2013 ISRN Combinatorics  
This paper is a contribution to the study of the automorphism groups of 2-(v,k,1) designs. Let 𝒟 be 2-(v,11,1) design and G≤ Aut(D) a block transitive and a point primitive.  ...  If G is unsolvable, then Soc(G), the socle of G, is not F42(q2).  ...  Acknowledgments This work is supported by the NNSFC (Grant no. 11271208) and the Scientific Reaserch Fund of Heilonjiang Provincial Education Departement (Grant no. 11553116).  ... 
doi:10.1155/2013/436987 fatcat:dy5u6jb3ebgt7cbwxe6ca36dg4

On block-transitive 2 - (ν, 4, 1) designs

Li Huiling
1995 Journal of combinatorial theory. Series A  
In [2] Camina and Siemons classified 2 --(v, 4, l) designs with a block-transitive, solvable group of automorphisms. They also proved some results about block-transitive 2-(v, k, 1) designs.  ...  If a block-transitive group of automorphisms of a 2 -(v, 4, 1) design is unsolvable, then it is flag-transitive. Reference [2] is an interesting paper.  ... 
doi:10.1016/0097-3165(95)90109-4 fatcat:btlzypprp5gbhezc74wqf7nwj4

On primitivity and reduction for flag-transitive symmetric designs

Eugenia O'Reilly Regueiro
2005 Journal of combinatorial theory. Series A  
a biplane admits a primitive, flag-transitive automorphism group of affine type.  ...  We present some results on flag-transitive symmetric designs. First we see what conditions are necessary for a symmetric design to admit an imprimitive, flag-transitive automorphism group.  ...  I also thank the referees, for their very helpful comments and suggestions, such as the argument for v = 247 at the end of the proof of Theorem 1, and the (96,20,4) imprimitive, flag-transitive example  ... 
doi:10.1016/j.jcta.2004.08.002 fatcat:mvch2ybgurfdnap47o7tpoedhu

Point Regular Normal Subgroups of Flag Transitive Automorphism Groups of 2-Designs

Paul-Hermann Zieschang
1996 Advances in Mathematics  
A point regular normal subgroup of a flag transitive automorphism group of a 2-design is solvable. Proof.  ...  Let D be a 2-design on v points with block size k, and let G be a flag transitive automorphism group of D. Let H be a one-point stabilizer of G, and let S be a subgroup of G such that H<S<G.  ...  Now, by the first equation of Lemma 5.1(i), p e =k p (q f &2), and the second equation of Lemma 5.2(ii) yields q f =k p +2. Thus, we have p e =k 2 p , 2 | e, and q f = p eÂ2 +2. Proof.  ... 
doi:10.1006/aima.1996.0049 fatcat:jyolqpkhlfeqpezppouaywoglq

Block-transitive and point-primitive 2-(v,k,2) designs with sporadic socle [article]

Xiaohong Zhang, Shenglin Zhou
2016 arXiv   pre-print
The purpose of this paper is to classify all pairs (D, G), where D is a non-trivial 2-(v, k, 2) design, and G≤ Aut(D) acts transitively on the set of blocks of D and primitively on the set of points of  ...  We prove that there exists only one such pair (D, G) in which D is a 2-(176,8,2) design and G=HS, the Higman-Sims simple group.  ...  Acknowledgements Thanks to Professor Sanming Zhou at University of Melbourne for corrections and some useful discussion which lead to the improvement of the paper.  ... 
arXiv:1603.07405v1 fatcat:nm2erjgj7ng7nortt5b5m734ri

A family of 4-designs with block size 9

Jürgen Bierbrauer
1995 Discrete Mathematics  
We construct a family of simple 4-designs with parameters 4-(2:+ 1,9, 84), (f, 6)= 1, f>~5.  ...  The G-orbit containing X forms a design 3-(q+l,k, 2) with 2= By definition each block is stabilized by a subgroup $3 of G. k(k-1)(k-2)/[K[. [] B is isomorphic to S 3. Proof.  ...  By this we mean a family {~i}, i=1,2 .... of designs with parameters t-(vi,k,2) where v~ ~ and such that none of the ~ is the complete design. When t and k are given, we wish to minimize 2.  ... 
doi:10.1016/0012-365x(94)00192-l fatcat:7nrxfdl5mvamtkgcj644bhdmfy

Flag-transitive block designs and finite exceptional simple groups of Lie type [article]

Seyed Hassan Alavi
2020 arXiv   pre-print
In this article, we study 2-designs with (r,λ)=1 admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type.  ...  We obtain four infinite families of such designs and provide some examples in each of these families.  ...  Praeger and Alice Devillers for supporting his visit to UWA (The University of Western Australia) during July-September 2019.  ... 
arXiv:1908.05831v4 fatcat:acglmx2anrcs7ii2owoclz2y64
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