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 admitting
we 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): 
Here
and
, 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.