Block-Transitive ( ) v k − 4 , , 4 Designs and Ree Groups

This article is a contribution to the study of the automorphism groups of ( ) v k − 4 , ,λ designs. Let ( ) = ,    be a non-trivial ( ) q k − + 3 4 1, ,4 design where n q + = 2 1 3 for some positive integer n ≥ 1 , and ( ) G Aut ≤  is block-transitive. If the socle of G is isomorphic to the simple groups of lie type ( ) G q 2 2 , then G is not flag-transitive.

, with the properties that each block is incident with k points, and each t-subset of  is incident with λ blocks.A flag of  is an incident point-block pair ( ) , x B with x is incident with B, where B ∈  .We consider automorphisms of  as pairs of permutations on  and  which preserve incidence structure.We call a group ( )

G Aut
≤  of automorphisms of  flag-transitive (respectively block-transitive, point t-transitive, point t-homogeneous) if G acts transitively on the flags (respectively transitively on the blocks, t-transitively on the points, t-homogeneously on the points) of  .For short,  is said to be, e.g., flag-transitive if  admits a flag-transitive group of automorphisms.
For historical reasons, a ( ) Recently, Huber (see [2]) completely classified all flag-transitive Steiner t-designs using the classification of the 2-transitive groups.Hence the determination of all flag-transitive and block-transitive t-designs with 2 λ ≥ has remained of particular interest and has been known as a long-standing and still open problem.
The present paper continues the work of classifying block-transitive t-designs.We discuss the block-transitive designs and Ree groups.We get the following result: Main Theorem.Let ( ) for some positive integer 1 n ≥ , and G q , then G is not flag-transitive.
The second section describes the definitions and contains several preliminary results about flag-transitivity and t-designs.In 3 Section, we give the proof of the Main Theorem.

Preliminary Results
The Ree groups ( ) G q form an infinite family of simple groups of Lie type, and were defined in [3] as subgroups of ( ) 7, GL q .Let ( ) GF q be finite field of q elements, where 2 1 for some positive integer ( ) G q is a group of order ( )( ) G q is a group of automorphisms of Steiner ( ) design and acts 2-transitive on 3 1 q + points (see [7]).Here we gather notation which are used throughout this paper.For a t-design ( ) Then G is block-transitive and the following cases hold: 1) , where x ∈  ; 2) , where B ∈  ; 3) ( ) , where x B ∈ .

Proof of the Main Theorem
Suppose that G acts flag-transitively on ( ) Then G is block-transitive and point-transitive.Since ( ) ( ) , we may assume that :

( )
x GF q ∈ and α is an automorphism of field ( ) + is odd, and m α = , then | m f .Obviously, ( )( ) fixes three different points of  , then g must fix at least four points in  .
There exist h P ∈ such that h z y = , where for all h ghg P − − ∈ .On the other hand, .
We get that C is transitive on , so ) B .We have . Therefore T acts also flag-transitively on ( ) ( )( ) Since G acts flag-transitively on ( ) design, then G is point-transitive.By Lemma 1(1), we get ( )( ) ( ) Again by Corollary 1, ( ) This is impossible.This completes the proof the Main Theorem.
where  denotes a set of points, v =  , and  a set of blocks, b =  r denotes the number of blocks through a given point, x G denotes the stabilizer of a point x ∈  and B G the setwise stabilizer of a block B ∈  .We define xB x B G G G =  .For integers m and n, let ( ) , m n denotes the greatest common divisor of m and n, and | m n if m divides n.Lemma 1. ([2]) Let G act flag-transitively on Investigating t-designs for arbitrary λ , but large t, Cameron and Praeger proved the following result: , ,t v k λ −design with 1 λ = is called a Steiner t-design (sometimes this is also known as a Steiner system).If t k v < < holds, then we speak of a non-trivial Steiner t-designs.