Block-transitive automorphism groups on 2-designs with block size 4
Introduction
Definition 1.1 For positive integers and , a - design is an incidence structure satisfying the following properties: is a set of elements, called points, is a set of
-subsets of , called blocks, Every -subset of is contained in exactly blocks.
Since all the blocks have the same size , it follows that each point belongs to the same number of blocks.
An automorphism of is a permutation of the points which also permutes the blocks. The set of all automorphisms of with the composition of maps is a group, denoted by . Let . If is a primitive permutation group on the point set , then is called point-primitive, otherwise point-imprimitive. A flag of is a point–block pair with , and is called flag-transitive if it acts transitively on the set of flags. Block-transitivity is defined similarly. A set of blocks of an incidence structure is called a set of base blocks with respect to an automorphism group of if it contains exactly one block from each -orbit on the block set. In particular, if is a block-transitive automorphism group of , then any block is a base block of .
It follows from a result of Ray-Chaudhuri and Wilson [7] that a block-transitive automorphism group of a 4-design is 2-homogeneous, and hence primitive on points. Thus, for a block-transitive point-imprimitive -design, the parameter is at most 3. It was shown in [2] that for a block-transitive and point-imprimitive - design , if preserves a partition with either 2 parts or parts of size 2. It is an open question whether or not this better holds bound in general for 3-design. In 1987, Davies [4] proved that, for a given value of , the block size of a flag-transitive point-imprimitive -design is bounded. In 1989, Delandtsheer and Doyen [5] proved that, if is a block-transitive point-imprimitive - design, then (see Lemma 2.3). It follows immediately that, for a block-transitive point-imprimitive - design, the number of points cannot be too large relative to the block size . Thus, if is a - design with a block-transitive and point-imprimitive automorphism group , then . The upper bound for 2-designs is attained has been almost completely answered in [2] when . The classification is an open problem for these values of . In this paper, we give a complete classification of - designs admitting a block-transitive point-imprimitive automorphism group.
We now state the first result of this paper:
Theorem 1 Let be a - design, and be a block-transitive point-imprimitive automorphism group of . Then and
Remark 1 The notation means that and there are exactly pairwise non-isomorphic such - designs. The base block and automorphism group of each design are given in Table 2.
For the point-primitive case, most research on block-transitive - designs hitherto has concerned the affine and almost simple types. However, there are very few studies that fully explore the block-transitive point-primitive - designs of product type. Recently, Zhan et al. determine all flag-transitive point-primitive - designs of product type in [9]. Focusing on 2- designs, we analyse the case in which the automorphism group is block-transitive point-primitive of product type and get the following:
Theorem 2 Let be a - design with a block-transitive point-primitive automorphism group . Assume that has a product action on the set . Then , and is a unique - design, a unique - design or a unique - design.
Section snippets
Preliminaries
Our notation and terminology are standard and can be found in [3], [6] for design theory and in [8] for group theory.
Lemma 2.1 The parameters and of a -design satisfy the following conditions: . .[3, 1.2, 1.9]
Let denote the stabilizer of a point , and the setwise stabilizer of a block .
Lemma 2.2 Let be a - design with a block-transitive automorphism group . Then , for all non-trivial subdegrees of . Moreover, divides .
Proof Let be a block of containing the point .
Proofs
In this section, we always assume that is a - design with a block-transitive automorphism group . In Section 3.1, we determine all pairs where is point-imprimitive group. In Section 3.2 we investigate the case which is point-primitive group of product type.
Declaration of Competing Interest
The authors declare that there is no conflict of interest in this paper.
Acknowledgements
The authors would like to thank referees for providing them helpful and constructive comments and suggestions, which led to the improvement of the article. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11801174, 11961026).
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