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Geometric groups of second order and related combinatorial structures
Journal of Combinatorial Designs ( IF 0.5 ) Pub Date : 2019-12-06 , DOI: 10.1002/jcd.21697
Andrew Woldar 1
Affiliation  

In 1977, D. Betten defined a geometric group to be a permutation group ( G , Ω ) such that G = Aut ( R ) for some hypergraph R on Ω . In this paper, we extend Betten's notion of a geometric group to what we call a geometric group of second order. By definition, this is a permutation group for which G = Aut ( R ) for some set R = { R 1 , R 2 , , R d } of hypergraphs on Ω . Our main focus will be on permutation groups that are geometric of second order but not geometric. Within this small class of groups one finds the projective groups P G L ( 2 , 8 ) , P Γ L ( 2 , 8 ) and the affine groups A G L ( 1 , 8 ) , A Γ L ( 1 , 8 ) . Our investigations, which are based primarily on these four groups, lead us to consider some familiar combinatorial structures (eg, Fano plane and affine design) in a less familiar context (overlarge sets of Steiner systems).

中文翻译:

二阶几何群和相关的组合结构

1977年,D。Betten将几何组定义为置换组 G Ω 这样 G = ut [R 对于一些超图 [R Ω 。在本文中,我们将Betten的几何群的概念扩展到了所谓的二阶几何群。根据定义,这是一个置换组 G = ut [R 对于一些集 [R = { [R 1个 [R 2 [R d } 上的超图 Ω 。我们的主要重点将放在二阶几何而不是几何的置换组上。在这一小类群体中,人们找到了投射群体 P G 大号 2 8 P Γ 大号 2 8 和仿射团体 一种 G 大号 1个 8 一种 Γ 大号 1个 8 。我们的调查主要基于这四个组,这使我们在不太熟悉的环境(大量Steiner系统)中考虑了一些熟悉的组合结构(例如Fano平面和仿射设计)。
更新日期:2019-12-06
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