Advances in Pure Mathematics
Vol.06 No.05(2016), Article ID:65362,4 pages
10.4236/apm.2016.65022
Block-Transitive Designs and Ree Groups
Shaojun Dai1, Ruihai Zhang2
1Department of Mathematics, Tianjin Polytechnic University, Tianjin, China
2Department of Mathematics, Tianjin University of Science and Technology, Tianjin, China
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 27 February 2016; accepted 4 April 2016; published 7 April 2016
ABSTRACT
This article is a contribution to the study of the automorphism groups of designs. Let
be a non-trivial
design where
for some positive integer
, and
is block-transitive. If the socle of G is isomorphic to the simple groups of lie type
, then G is not flag-transitive.
Keywords:
Flag-Transitive, Block-Transitive, t-Design, Ree Group
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.
Cite this paper
Shaojun Dai,Ruihai Zhang, (2016) Block-Transitive 4-(v,k,4) Designs and Ree Groups. Advances in Pure Mathematics,06,317-320. doi: 10.4236/apm.2016.65022
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