Received 27 February 2016; accepted 4 April 2016; published 7 April 2016
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
with x is incident with B, where
. We consider automorphisms of
as pairs of permutations on
and
which preserve incidence structure. We 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-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 design with
is called a Steiner t-design (sometimes this is also known as a Steiner system). If
holds, then we speak of a non-trivial Steiner t-designs.
Investigating t-designs for arbitrary, but large t, Cameron and Praeger proved the following result:
Theorem 1. ( [1] ) Let be a
design. If
acts block-transitively on
, then
, while if
acts flag-transitively on
, then
.
Recently, Huber (see [2] ) completely classified all flag-transitive Steiner t-designs using the classification of the finite 2-transitive permutation groups. Hence the determination of all flag-transitive and block-transitive t-designs with 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 be a non-trivial
design, where
for some positive integer
, and
is block-transitive. If
, the socle of G, is
, 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.
2. Preliminary Results
The Ree groups form an infinite family of simple groups of Lie type, and were defined in [3] as subgroups of
. Let
be finite field of q elements, where
for some positive integer
(in particular,
). Let Q is a Sylow 3-subgroup of G, K is a multiplicative group of
and
is a group of order
(see [4] - [6] ). Hence
is a group of automorphisms of Steiner
design and acts 2-transitive on
points (see [7] ).
Here we gather notation which are used throughout this paper. For a t-design with
, let r denotes the number of blocks through a given point,
denotes the stabilizer of a point
and
the setwise stabilizer of a block
. We define
. For integers m and n, let
denotes the greatest common divisor of m and n, and
if m divides n.
Lemma 1. ( [2] ) Let G act flag-transitively on design
. Then G is block-transitive and the following cases hold:
1), where
;
2), where
;
3), where
.
Lemma 2. ( [8] ) Let is a non-trivial
design. Then
Lemma 3. ( [8] ) Let is a non-trivial
design. Then
1);
2).
Corollary 1. Let is a non-trivial
design. If
, Then
.
Proof. By Lemma 2, we have. If
, then
Hence
We get
3. Proof of the Main Theorem
Suppose that G acts flag-transitively on design and
. Then G is block-transitive and point-transitive. Since
, we may assume that
and
by Dedekind’s theorem, where
,
and a is an automorphism of field
. Let
,
is odd, and
, then
. Obviously,
.
First, we will proof that if fixes three different points of
, then g must fix at least four points in
.
Suppose that,
,
. Let P is a normal Sylow 3-subgroup of
. Then
is transitive on. By
, we have
. Hence P acts regularly on
. There exist
such that
, where for all
. Since
,
and P is a normal
Sylow 3-subgroup of, we have
. On the other hand,
So, that is
. Hence
. We get that C is transitive on
. Hence
. By
, we have
. Note that
, so
. Hence
. It follows that
. This means that g must fix at least four points in
.
Now, we can continue to prove our main theorem. Obviously, fixes three points of
which are
. Then
. Hence
must fix at least five points in
. Since G acts block-transitively on
design, we can find four blocks, let
,
,
and
, containing four points which is fixed by a. If a exchange
,
,
and
, then
which is impossible. Thus a must fix
,
,
and. We have
. Therefore T acts also flag-transitively on
design. We may assume
and
.
Since G acts flag-transitively on design, then G is point-transitive. By Lemma 1(1), we get
Again by Lemma 3(2) and Lemma 1(3),
Thus
By Lemma 2,
Again by Corollary 1,
This is impossible.
This completes the proof the Main Theorem.
[1] Cameron, P.J. and Praeger, C.E. (1992) Block-Transitive t-Designs, II: Large t. In: De Clerck, F., et al., Eds., Finite geometry and Combinatorics, Deinze. (1993) London Math. Soc. Lecture Note Series, Vol. 191, Cambridge Univ. Press, Cambridge, 103-119.
[2] Huber, M. (2009) Flag-Transitive Steiner Designs. Birkhauser Basel, Berlin, Boston.
http://dx.doi.org/10.1007/978-3-0346-0002-6
[3] Ree, R. (1961) A Family of Simple Groups Associated with the Simple Lie Algebra of Type (G2). American Journal of Mathematics, 83, 432-462.
http://dx.doi.org/10.2307/2372888
[4] Kleidman, P.B. (1988) The Maximal Subgroups of the Chevally Groups 2G2(q) with q Odd, the Ree Groups 2G2(q), and Their Automorphism Groups. Journal of Algebra, 117, 30-71.
http://dx.doi.org/10.1016/0021-8693(88)90239-6
[5] Liu, W.J., Zhou, S., Li, H. and Fan, X. (2004) Finite Linear Spaces Admitting a Ree Simple Group. European Journal of Combinatorics, 25, 311-325.
http://dx.doi.org/10.1016/j.ejc.2003.10.001
[6] Dai, S.J. and Zhang, R.H. (2013) The Designs with Block-Transitive Automorphism. Ars Combinatoria, 110, 15-21.
[7] Dixon, J.D. and Mortimer, B. (1996) Permutation Groups. Springer Verlag, Berlin.
http://dx.doi.org/10.1007/978-1-4612-0731-3
[8] Shen, H. (1990) The Theory of Combinatorial Design. Shanghai Jiaotong Univ. Press, Shanghai.