Elsevier

Applied Mathematics and Computation

Volume 242, 1 September 2014, Pages 109-115
Applied Mathematics and Computation

Block-transitive 2-(v,k,1) designs and the twisted simple groups 2E6(q)

https://doi.org/10.1016/j.amc.2014.05.027Get rights and content

Abstract

This article is a contribution to the study of block-transitive automorphism groups of 2-(v,k,1) block designs. Let D be a 2-(v,k,1) design admitting a block-transitive, point-primitive but not flag-transitive group G of automorphisms. Let kr=(k,v-1) and q=pf for prime p. In this paper we prove that if G and D are as above and q> (3(krk-kr+1)f)1/3 then G does not admit a twisted simple group 2E6(q) as its socle.

Introduction

A 2-(v,k,1) design D=(P,B) is a pair consisting of a finite set P of v points and a collection B of k-subsets of P, called blocks, such that each 2-subset of P is contained in exactly one block. We will always assume that 2<k<v.

Recall that an automorphism of a 2-(v,k,1) design D is a permutation of the set P of points which maps blocks to blocks. The set of all automorphisms is called the automorphism group Aut(D) of D, a subgroup of Sym(P). Let GAut(D), then G is said to be block transitive on D if G is transitive on B, and is said to be point transitive (point primitive) on D if G is transitive (primitive) on P. A flag of D is a pair consisting of a point and a block through that point. Then G is flag transitive on D if G is transitive on the set of flags.

In 1990, a six-person team [4] classified the pairs (G,D) where G is a flag-transitive automorphism group of D, with the exception of those in which G is a one-dimensional affine group. In this paper we contribute to the classification of designs which have an automorphism group transitive on blocks. It follows from a result of Block [2] that a block-transitive automorphism group of a 2-(v,k,1) design is transitive on points. In [7] it is shown that the study of block-transitive 2-(v,k,1) designs can be reduced to three cases, distinguishable by properties of the action of G on the point set P: that in which G is of affine type in the sense that it has an elementary abelian transitive normal subgroup; that in which G is almost simple, in the sense that G has a simple nonabelian transitive normal subgroup T whose centralizer is trivial, so that TGAutT; and that in which G has an intransitive minimal normal subgroup. Much work is needed to achieve this classification, see [5], [6], [7], [8], [9], [12]. W. Liu et al. have studied the special case where G=TSoc(G) is any finite group of Lie type of Lie rank 1 acting block-transitively on a design in [15], [16], [17], [18]. Here we focus on the second case, that is classifying 2-(v,k,1) designs with a block-transitive automorphism group of almost simple type under the conditions that G is point-primitive but not flag-transitive. We prove the following theorem:

Theorem 1

Let D be a 2-(v,k,1) design admitting a block-transitive, point-primitive but not flag-transitive automorphism group G. Let kr=(k,v-1),q=pf for some prime p and positive integer f. If q> (3(krk-kr+1)f)1/3 then Soc(G)2E6(q).

The assumption q>(3(krk-kr+1)f)1/3 is necessary for the proof of Theorem 1. Our proof depends on the result of Liebeck and Saxl [14] about the classification of maximal subgroups of T=Soc(G), and the properties of the lengths of the suborbits of T, given in Section 3. We shall continue this work in a forthcoming paper dealing with q small and using different methods.

Our paper is organized as follows: In Section 2 we collect some preliminary results and in Section 3 we use them to prove Theorem 1.

Section snippets

Preliminary results

Let D be a 2-(v,k,1) design defined on the point set P, and suppose that G is an automorphism group of D that acts transitively on blocks. For a 2-(v,k,1) design, as usual, b denotes the number of blocks and r denotes the number of blocks through a given point. If B is a block, GB denotes the setwise stabilizer of B in G and G(B) is the pointwise stabilizer of B in G. Also, GB denotes the permutation group induced by the action of GB on the points of B, and so GBGB/G(B).

For the basic notions

Proof of Theorem 1

We assume throughout this section that q=pf for p a prime integer and f a positive integer. If n is a positive integer, then np denotes the p-part of n and np denotes the p-part of n. In other words, np=pt where pt|n but pt+1n, and np=n/np.

Following Fang-Li (see [11]), we shall use the following parameters of 2-(v,k,1) designs:kv=(k,v),kr=(k,r)=(k,v-1),bv=(b,v),br=(b,r)=(b,v-1).It is easy to check thatk=kvkr,b=bvbr,v=kvbvandr=krbr.

Proposition 3.1

Let D and G satisfy the conditions of Theorem 1, T=Soc(G) and

Acknowledgment

The author would like to thank the referees for their valuable comments and suggestions on this paper.

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This work partially supported by the National Natural Science Foundation of China (Grant Nos. 11071081, 11271208, 61273093) and the Natural Science Foundation of Zhejiang Province (Grant No. LY12A01004).

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