ABSTRACT

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.

Keywords:Flag-Transitive, Combinatorial Design, Permutation Group, Affine Group, 3-Homogeneous Permutation Groups

1. Introduction

For positive integers and, we define a design to be a finite incidence structure, where denotes a set of points, and a set of blocks, , with the properties that each block is incident with k points, and each t-subset of is incident with blocks. A flag of is an incident point-block pair, that is and such that. We consider automorphisms of as pairs of permutations on X and B which preserve incidence, and call a group 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-homogeneous on the points) of. 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 flagtransitive Steiner t-designs, Michael Huber has done the classification (See [3] -[7] ). But only a few people have discussed the case of flag-transitive t-designs where and.

In this paper, we may study a kind of flag-transitive designs with. We may consider this problem by making use of the classification of the finite 3-homogeneous permutation groups to study flag-transitive designs. Our main result is:

Theorem: There are no non-trivial designs admitting a flag transitive group of automorphisms.

2. Preliminary Results

Lemma 2.1. (Huber M [4] ) Let be a design with .If acts flag-transitively on, then G also acts point 2-transitively on.

Lemma 2.2. (Cameron and Praeger [8] ). Let be a design with. Then the following holds:

(1) If acts block-transitively on, then also acts point -homogeneously on;

(2) If acts flag-transitively on, then also acts point -homogeneously on.

Lemma 2.3. (Huber M [9] ) Let be a design. If acts flag-transitively on, then , for any , the division property holds.

Lemma 2.4. Let be a design. Then the following holds:

(1);

(2);

(3) For a design is also an design, where.

(4) In particular, if t = 6, then

Lemma 2.5. (Beth T [10] ) If is a non-trivial design, then

Lemma 2.6. (Wei J L [11] ) If is a design, then

In this case, when, we deduce from Lemma 2.6 the following upper bound for the positive integer.

Corollary 2.7. Let be a non-trivial design, then

.

Proof: By Lemma 2.6, when, we have, then

.

Remark 2.8. Let be a non-trivial design with. If acts flagtransitively on, then by Lemma 2.2 (1), acts point 3-homogeneously and in particular point 2-transitively on. Applying Lemma 2.4 (2) yields the equation

where and are two distinct points in and is a block in. If then

.

Corollary 2.9 Let be a design, then

For each positive integers,.

Let G be a finite 3-homogeneous permutation group on a set X with. Then is either of

(A) Affine Type:

contains a regular normal subgroup which is elementary Abelian of order.If we identify with a group of affine transformations

Of, where and, then particularly one of the following occurs:

(1)

(2);

(3);

or

(B) Almost Simple Type: contains a simple normal subgroup, and. In particular, one of the following holds, where and are given as follows:

(1)

(2)

(3)

(4).

3. Proof of the Main Theorem

Let be a non-trivial design, acts flag-transitively on, by lemma 2.2, is a finite 3-homogeneous permutation group. For is a non-trivial design, then We will prove by contradiction that cannot act flag-transitively on any non-trivial design.

3.1. Groups of Automorphisms of Affine Type

Case (1):

If, then Lemma 2.5 yields, a contradiction to. For, Corollary 2.7 implies. Thus By Lemma 2.4 we have

for each values of, we have

but is a positive integer, thus On the other hand, we have , those are contradicting to Lemma 2.3.

Case (2):.

Here For, we have, already ruled out in Case (1). So we may assume that. Any six distinct points being non-coplanar in, they generate an affine subspace of dimension at least 3. Let be the 3-dimensional vector subspace spanned by the first three basis vectors of the vector space. Then the point-wise stabilizer of in (and therefore also in) acts point-transitively on. Let and be the two blocks which are incident with the 6-subset , If the block contains some point of, then contains all points of, and so, this yields, a contradiction to Lemma 2.6. Hence and. On the other hand, for is a flag-transitive 6-design admittingwe deduce from [[12] , prop.3.6 (b)] the necessary condition that must divide, and hence it follows for each respective value of that, contradicting our assumption.

Case (3):

For, we have, by Corollary 2.7. By Lemma 2.4 and Lemma 2.3, we have.

3.2. Groups of Automorphisms of Almost Simple Type

Case (1):

Since is non-trivial with, we may assume that. Then is 6-transitive on, and hence is -transitive, this yields containing all of the -subset of. So is a trivial design, a contradiction.

Case (2):

Hereand, so with and. We may again assume that.

We will first assume that. Then, by Remark 2.8, we obtain

(1)

In view of Lemma 2.6, we have

(2)

It follows from Equation (1) that

(3)

If we assume that, then obviously

and hence

In view of inequality (2), clearly, this is only possible when. In particular, 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, then the few remaining possibilities for can easily be ruled out by hand using Equation (1), Inequality (2), and Corollary 2.9.

Now, let us assume that. We recall that, and will distinguish in the following the case

First, let. We define with of order

induced by the Frobenius automorphism. Then, by Dedekind’s law, we can write

Defining, it can easily be calculated that, and has precisely distinct fixed points (cf. e.g., [[13] Ch. 6.4, Lemma 2]). As, we have therefore that for a flag fixed with by the definition of designs. On the other hand, every element of either fixes block, or commute block with block, thus the index. Clearly

Hence, we have

where. Thus, if we assume that acts already flag-transitively on, then we obtain

Then either and acts on flag-transitively, that is the case when; or and has exactly two orbits of equal length on the sets of flags. Then, proceeding similarly to the case for each orbit on the set of the flags, we have that

(4)

Using again

(5)

We obtain

(6)

If we assume that, then again

(7)

and thus

but this is impossible. The few remaining possibilities for can again easily be ruled out by hand.

Now, let then, clearly, and we have. If we assume that is the subgroup of for a flag, then we have and as clearly, we can apply Equation. Thus, must also be flagtransitive, which has already been considered. Therefore, we assume that is not the subgroup of. Let

be a prime divisor of. As the normal subgroup of index

has precisely distinct fix points, we have for a flag fixed with by the definition of designs. It can then be deduced that for some Since if we assume for that there exists a further prime divisor of with, then

and are both subgroups of by the flag-transitivity of, and hence, a contradiction. Furthermore, as is not the subgroup of

. We may, by applying Dedekind’s law, assume that

Thus, by Remark 2.8, we obtain

More precisely:

(A) if,

(B) if,

As far as condition (A) is concerned, we may argue exactly as in the earlier case. Thus, only condition (B) remains. If is a power of 2, then Remark 2.8 gives

with. In particular, must divide, and we may proceed similarly as in the case, yielding a contradiction.

The case may be treated as the case.

Case (3):

By Corollary 2.7, we get for or 12, and or 8 for or 24, and the very small number of cases for can easily be eliminated by hand using Corollary 2.9 and Remark 2.8.

Case (4):

As in case (3), for, we have 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.

Acknowledgements

The authors thank the referees for their valuable comments and suggestions on this paper.

References