Flag-Transitive 6-( v , k , 2 ) Designs

The automorphism group of a flag-transitive 6–(v, k, 2) design is a 3-homogeneous permutation group. Therefore, using the classification theorem of 3–homogeneous permutation groups, the classification of flag-transitive 6-(v, k,2) designs can be discussed. In this paper, by analyzing the combination quantity relation of 6–(v, k, 2) design and the characteristics of 3-homogeneous permutation groups, it is proved that: there are no 6–(v, k, 2) designs D admitting a flag transitive group G ≤ Aut (D) of automorphisms.


Introduction
∈ .We consider automorphisms of D as pairs of permutations on X and B which preserve incidence, and call a group ( ) of automorphisms of D 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-homogeneous on the points) of D .It is a different problem in Combinatorial Maths how to construct a design with given parameters.In this paper, we shall take use of the automorphism groups of designs to find some new designs.
In recent years, the classification of flag-transitive Steiner 2-designs has been completed by W. M. Kantor (See [1]), F. Buekenhout, A. De-landtsheer, J. Doyen, P. B. Kleidman, M. W. Liebeck, J. Sax (See [2]); for flag-transitive Steiner t-designs ( ) 2 6 t < ≤ , Michael Huber has done the classification (See [3]- [7]).But only a few people have discussed the case of flag-transitive t-designs where 3 t > and 1 λ > .In this paper, we may study a kind of flag-transitive designs with 2 λ = .We may consider this problem by making use of the classification of the finite 3-homogeneous permutation groups to study flag-transitive ( )
acts flag-transitively on D , then G also acts point 2-transitively on D .
Lemma 2.2.(Cameron and Praeger [8]).Let ( ) Then the following holds: (1) If ( ) acts block-transitively on D , then G also acts point acts flag-transitively on D , then G also acts point ( ) (2) acts flagtransitively on D , then by Lemma 2.2 (1), G acts point 3-homogeneously and in particular point 2-transi- tively on D .Applying Lemma 2.4 (2) yields the equation ( ) where x and y are two distinct points in X and 1 For each positive integers, s t ≤ .Let G be a finite 3-homogeneous permutation group on a set X with 4 X ≥ .Then G is either of (A) Affine Type: G contains a regular normal subgroup T which is elementary Abelian of order 2 d v = .If we identify G with a group of affine transformations , then particularly one of the following occurs: (1) In particular, one of the following holds, where N and v X = are given as follows: (

Proof of the Main Theorem
acts flag-transitively on D , by lemma 2.2, G is a finite 3-homogeneous permutation group.For D is a non-trivial We will prove by contradiction that ( ) cannot act flag-transitively on any non-trivial

Groups of Automorphisms of Affine Type
Case (1): , they generate an affine subspace of dimension at least 3. Let ε be the 3-dimensional vector subspace spanned by the first three basis vectors 1 2 3 , , e e e of the vector space ( ) Then the point-wise stabilizer of ε in

Groups of Automorphisms of Almost Simple Type
Case (1): A is 6-transitive on X , and hence G is k -transitive, this yields D containing all of the k -subset X .So D is a trivial design, a contra- diction.
Case (2): e PLS q v q q p =+ = > Here ( ) N PLS q v q q p = =+ ≥ and and a de .We may again assume that 1 8 v q = + ≥ .We will first assume that N G = .Then, by Remark 2.8, we obtain In view of Lemma 2.6, we have ( ) ( )( ) It follows from Equation ( 1) that If we assume that 21 k ≥ , then obviously In view of inequality (2), clearly, this is only possible when ( )

2, 1
xB PSL q = .In particular, q has not to be even.But then the right-hand side of Equation ( 1) is always divisible by 16 but never the left-hand side, a contradiction.If 21 k < , then the few remaining possibilities for k can easily be ruled out by hand using Equa- tion (1), Inequality (2), and Corollary 2.9.Now, let us assume that ( ) . We recall that 7 e q p = ≥ , and will distinguish in the following the case 3, 2, and 3.
. Thus, if we assume that ( ) acts already flag-transitively on D , then we obtain ( )

2,
PSL q acts on D flag-transi- tively, that is the case when N G = ; or ( ) 2, PSL q has exactly two orbits of equal length on the sets of flags.Then, proceeding similarly to the case N G = for each orbit on the set of the flags, we have that Using again ( ) ( )( ) We obtain If we assume that 21 k ≥ , then again but this is impossible.The few remaining possibilities for 21 k < can again easily be ruled out by hand.Now, let 2, p = then, clearly , and we have

2,
PSL q must also be flagtransitive, which has already been considered.Therefore, we assume that α τ is not the subgroup of ( )

2, B P L q
Σ by the flag-transitivity of ( ) , and hence ( ) , a contradiction.Furthermore, as α τ is not the subgroup of ( ) We may, by applying Dedekind's law, assume that ( ) ( ) As far as condition (A) is concerned, we may argue exactly as in the earlier case N G = .Thus, only condition (B) remains.If e is a power of 2, then Remark 2.8 gives with a e .In particular, a must divide 1 0B G , and we may proceed similarly as in the case N G = , yielding a contradiction.
The case 3 p = may be treated as the case

For
positive integers t k v ≤ ≤ and λ , we define a X denotes a set of points, X v = and Β a set of blocks, b Β = , with the properties that each block B ∈ Β is incident with k points, and each t-subset of X is incident with λ blocks.A flag of D is an incident point-block pair, that is x X ∈ and B ∈ Β such that ( ) , x B I Almost Simple Type: G contains a simple normal subgroup N , and

2
SL d (and therefore also in G ) acts point-transitively on \ V ε .Let 1 B and B′ be the two blocks which are incident with the 6-subset , , , e e e e e e e + + , If the block 1 B B′  contains some point α of \ V ε , then 1 On the other hand, for D is a flag-transitive 6-design admitting

we deduce from Lemma 2.6 the following upper bound for the positive integer k . Corollary 2.7. Let
order e and the very small number of cases for k can easily be eliminated by hand using Corollary 2.9 and Remark 2.8.= in view of Corollary 2.7, a contradiction since no 6-(12, 7, 2) design can exist by Corollary 2.9.This completes the proof of the Main Theorem.