Generating groups using hypergraphs
Generating groups using hypergraphs
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DOI：
10.1093/qmath/haw001
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发表时间：
2014-05
期刊：
影响因子：
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通讯作者：
Nick Gill;Neil I. Gillespie;A. Nixon;Jason Semeraro
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文献类型：
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作者：
Nick Gill;Neil I. Gillespie;A. Nixon;Jason Semeraro
对于大小为$n$的集合$\Omega$的4个子集的集合$\mathcal{B}$，我们引入了一个称为“洞稳定器”的不变量，它推广了基于lloyd的“15谜题”的Conway， Elkies和Martin的Mathieu群$M_{12}$的构造。证明了孔洞稳定器可以看作是一个客观偏群（Chermak意义上的）内的物体。我们用一个平凡的孔稳定剂对$(\Omega,\mathcal{B})$进行分类，并用$\lambda \leq 2$确定与$2$ - $(n,4,\lambda)$设计相关的所有孔稳定剂。
To a set $\mathcal{B}$ of 4-subsets of a set $\Omega$ of size $n$ we introduce an invariant called the `hole stabilizer' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd's `15-puzzle'. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs $(\Omega,\mathcal{B})$ with a trivial hole stabilizer, and determine all hole stabilizers associated to $2$-$(n,4,\lambda)$ designs with $\lambda \leq 2$.