Block-transitive 2- designs and the twisted simple groups ☆
Introduction
A 2- design =() is a pair consisting of a finite set of v points and a collection of k-subsets of , called blocks, such that each 2-subset of is contained in exactly one block. We will always assume that .
Recall that an automorphism of a 2- design is a permutation of the set of points which maps blocks to blocks. The set of all automorphisms is called the automorphism group of , a subgroup of Sym(). Let , then G is said to be block transitive on if G is transitive on , and is said to be point transitive (point primitive) on if G is transitive (primitive) on . A flag of is a pair consisting of a point and a block through that point. Then G is flag transitive on if G is transitive on the set of flags.
In 1990, a six-person team [4] classified the pairs where G is a flag-transitive automorphism group of , 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- design is transitive on points. In [7] it is shown that the study of block-transitive 2- designs can be reduced to three cases, distinguishable by properties of the action of G on the point set : 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 ; 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 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- 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 be a 2- design admitting a block-transitive, point-primitive but not flag-transitive automorphism group G. Let for some prime p and positive integer f. If then .
The assumption 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 , 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 be a 2- design defined on the point set , and suppose that G is an automorphism group of that acts transitively on blocks. For a 2- 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, denotes the setwise stabilizer of B in G and is the pointwise stabilizer of B in G. Also, denotes the permutation group induced by the action of on the points of B, and so .
For the basic notions
Proof of Theorem 1
We assume throughout this section that for p a prime integer and f a positive integer. If n is a positive integer, then denotes the p-part of n and denotes the -part of n. In other words, where but , and .
Following Fang-Li (see [11]), we shall use the following parameters of 2- designs:It is easy to check that Proposition 3.1 Let and G satisfy the conditions of Theorem 1, 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).