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

2001
*
Journal of combinatorial theory. Series A
*

Let G be a

doi:10.1006/jcta.2000.3067
fatcat:spphi2bw7ravxg5kcnmim5ijvm
*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*. ...##
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Solvable block transitive automorphism groups of 2−(v,5,1) designs

2003
*
Discrete Mathematics
*

Let G be a

doi:10.1016/s0012-365x(02)00763-x
fatcat:ul6rtui3sfcltgqdn2olb6lufu
*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*. ...##
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Unsolvable block transitive automorphism groups of 2-(v,k,1)(k=6,7,8,9) designs

2008
*
Discrete Mathematics
*

In this paper, we study

doi:10.1016/j.disc.2007.10.051
fatcat:7u5jnqktjnhgvk2boxpva5dvsa
*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*. ...##
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Block transitive 2-(v,k,1) designs and two-dimensional projective linear groups

2015
*
International Journal of Algebra
*

Let G act as a

doi:10.12988/ija.2015.528
fatcat:4cybp4qayvgwroobfkx6vexbwu
*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*,*v*−*1*). ... 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*. ...##
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Nonexistence of $2-(v, k, 1)$ designs admitting automorphism groups with socle $E_8(q)$

2015
*
Miskolc Mathematical Notes
*

In particular, the existence

doi:10.18514/mmn.2015.1473
fatcat:w4k7qonjyfdntp6cxjgqmcl6g4
*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. ...##
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The Ree groups 2G2(q) and 2−(v,k,1) block designs

2000
*
Discrete Mathematics
*

Let G act as a

doi:10.1016/s0012-365x(00)00050-9
fatcat:bcucw7iejfcqpbcgd4eszmteny
*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. ...##
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Automorphism groups of designs

1969
*
Mathematische Zeitschrift
*

If F is a flag-

doi:10.1007/bf01111409
fatcat:3xuznnemwncpjnmkssqtgpqf5a
*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*. ...##
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Block-transitive, point-primitive Steiner 3-designs
[article]

2021
*
arXiv
*
pre-print

This paper studies the long-standing open problem

arXiv:2112.00466v3
fatcat:r3z4iaxfwnaklbma4hzzrsal3m
*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. ...##
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The 2-(v,11,1) Designs with Unsolvable Block Transitive Automorphism

2013
*
ISRN Combinatorics
*

This paper is a contribution to the study

doi:10.1155/2013/436987
fatcat:dy5u6jb3ebgt7cbwxe6ca36dg4
*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). ...##
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On block-transitive 2 - (ν, 4, 1) designs

1995
*
Journal of combinatorial theory. Series A
*

In [

doi:10.1016/0097-3165(95)90109-4
fatcat:btlzypprp5gbhezc74wqf7nwj4
*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. ...##
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On primitivity and reduction for flag-transitive symmetric designs

2005
*
Journal of combinatorial theory. Series A
*

a biplane admits a primitive, flag-

doi:10.1016/j.jcta.2004.08.002
fatcat:mvch2ybgurfdnap47o7tpoedhu
*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 ...##
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Point Regular Normal Subgroups of Flag Transitive Automorphism Groups of 2-Designs

1996
*
Advances in Mathematics
*

A point regular normal subgroup

doi:10.1006/aima.1996.0049
fatcat:jyolqpkhlfeqpezppouaywoglq
*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. ...##
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Block-transitive and point-primitive 2-(v,k,2) designs with sporadic socle
[article]

2016
*
arXiv
*
pre-print

The purpose

arXiv:1603.07405v1
fatcat:nm2erjgj7ng7nortt5b5m734ri
*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. ...##
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A family of 4-designs with block size 9

1995
*
Discrete Mathematics
*

We construct a family

doi:10.1016/0012-365x(94)00192-l
fatcat:7nrxfdl5mvamtkgcj644bhdmfy
*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*. ...##
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Flag-transitive block designs and finite exceptional simple groups of Lie type
[article]

2020
*
arXiv
*
pre-print

In this article, we study

arXiv:1908.05831v4
fatcat:acglmx2anrcs7ii2owoclz2y64
*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. ...
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