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Additivity of affine designs
Journal of Algebraic Combinatorics ( IF 0.8 ) Pub Date : 2020-06-13 , DOI: 10.1007/s10801-020-00941-8
Andrea Caggegi , Giovanni Falcone , Marco Pavone

We show that any affine block design \(\mathcal{D}=(\mathcal{P},\mathcal{B})\) is a subset of a suitable commutative group \({\mathfrak {G}}_\mathcal{D},\) with the property that a k-subset of \(\mathcal{P}\) is a block of \(\mathcal{D}\) if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design \(\mathcal{D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal{D}\) that leave \(\mathcal P\) invariant. Whenever k is a prime p, \({\mathfrak {G}}_\mathcal{D}\) is an elementary abelian p-group.



中文翻译:

仿射设计的可加性

我们证明任何仿射块设计\(\ mathcal {D} =(\ mathcal {P},\ mathcal {B})\)是合适的交换组\({\ mathfrak {G}} _ \ mathcal的子集{D},\)具有以下性质:\(\ mathcal {P} \)k个子集是\(\ mathcal {D} \)的块,且仅当其k个元素的总和为零时。结果,任何仿射设计\(\ mathcal {D} \)的自同构组就是\({\ mathfrak {G}} _ \ mathcal {D} \)的自同构组,而剩下\(\ mathcal P \)不变。只要k是素数p\({\ mathfrak {G}} _ \ mathcal {D} \)是基本的阿贝尔p-群。

更新日期:2020-06-13
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