Infinite families of strictly cyclic Steiner quadruple systems
Infinite families of strictly cyclic Steiner quadruple systems
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DOI：
10.1016/0012-365x(89)90369-5
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发表时间：
1989-09
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通讯作者：
H. Siemon
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作者：
H. Siemon
u阶的Steiner四重系统SQS (v)是一对（v， B），其中v是包含υεΝ*个元素的集合，B是(4)个元素的子集，这些元素被称为块，因此v的每3个子集都包含在一个唯一的块中。H. Hanani[1]证明了SQS (v)存在的必要条件v= 2,4(6)也是充分的。在A. Hartman[2,3]和Lenz[5]的论文中，对Hanani的证明进行了简化。然而，如果我们需要一个SQS (v)来允许一个给定的自同构群，那么SQS (v)的存在性问题还没有完全解决，即使这个自同构群是v阶循环的。一个具有v阶循环自同构群C的SQS (v)称为循环的，记为CSQS (v)。如果CSQS (v)的任意四重元的稳定子等于恒等元（C的轨道都有长度v），我们称其为严格循环SQS (v)，记为sSQS (v)。在b[7]中，我们构建了sSQS（2.5º）。在本文中，我们将我们的构造扩展到sSQS (2p), p= 5(12)，前提是sSQS （2p）存在包含基四元组{0，i, 2i，½+ i}, i= 1,2，…，(v-2)/4，所有轨道在映射i→-i （mod v）下不变。在最近关于循环斯坦纳四重系统的论文列表中（参见[7]），我们必须加上Piotrowski[6]的论文，他在其工作的主要部分证明了以下定理：(i)一个具有2v阶二面体群D的SQS (v)存在自同构群，如果v= 0 (2), υ≠0 (3),υ≠0 (8),v≥4，并且对于v的任何素数p存在一个具有D₂p的SQS （2p）作为自同构群。（ii）对于所有素数p= 1(4)且p≤229，或p= 1(4)且p≠1,49（60）且p< 15000，存在自同构群A={(x→ax+ b| A, b∈Z，且gcd (A, v)= 1}的SQS （2p）。当p≠1(3)时，这个SQS （2p）有D₂p作为自同构群。
A Steiner Quadruple System SQS (v) of order u is a pair (V, B) where V is a set with υεΝ* elements, B a subset of (4) the elements of which are called blocks so that every 3-subset of V is contained in a unique block. H. Hanani [1] proved that the necessary condition v= 2, 4 (6) for the existence of a SQS (v) is also sufficient. In papers by A. Hartman [2, 3] and Lenz [5] Hanani's proof was simplified. If, however, we require a SQS (v) to allow a given automorphism group the problem of the existence of SQS (v) is not yet solved completely, even if the automorphism group is cyclic of order v. A SQS (v) with a cyclic automorphism group C, of order v is called cyclic, denoted CSQS (v). If the stabilizer of any quadruple of a CSQS (v) equals the identity (the orbits of C have all length v) we speak of a strictly cyclic SQS (v), denoting them sSQS (v). In [7] we constructed among other things sSQS (2.5 º). In this paper we will extend our construction to sSQS (2p), p= 5 (12) provided sSQS (2p) exists containing the base quadruples {0, i, 2i, ½+ i}, i= 1, 2,...,(v-2)/4 and all orbits invariant under the mapping i→-i (mod v). To the list of recent papers which deal with cyclic Steiner Quadruple Systems (cf.[7]) we have to add the dissertation by Piotrowski [6], who proved, in the main part of his work, the following theorems:(i) A SQS (v) with dihedral group D, of order 2v as automorphism group exists iff v= 0 (2), υ≠ 0 (3), υ≠ 0 (8), v≥ 4 and if for any prime divisor p of v there exists a SQS (2p) with D₂p as automorphism group.(ii) For all prime numbers p= 1 (4) and p≤ 229, or p= 1 (4) and p≠ 1, 49 (60) and p< 15000 there exists SQS (2p) with the automorphism group A={(x→ ax+ b| a, b∈ Z, and gcd (a, v)= 1}. In case p≠ 1 (3) this SQS (2p) has D₂p as an automorphism group.