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Maximal nonassociativity via nearfields
Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2019-11-18 , DOI: 10.1016/j.ffa.2019.101610
Aleš Drápal , Petr Lisoněk

We say that (x,y,z)Q3 is an associative triple in a quasigroup Q() if (xy)z=x(yz). It is easy to show that the number of associative triples in Q is at least |Q|, and it was conjectured that quasigroups with exactly |Q| associative triples do not exist when |Q|>1. We refute this conjecture by proving the existence of quasigroups with exactly |Q| associative triples for a wide range of values |Q|. Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums.



中文翻译:

通过近场实现最大非缔合

我们说 Xÿž3 是准群中的关联三元组 如果 Xÿž=Xÿž。很容易证明,Q中的三联体数目至少为||,并且推测准群与 || 关联三元组不存在时 ||>1个。我们通过证明具有|| 多种值的关联三元组 ||。我们的主要工具是二次Dickson近场和Weil结合二次字符和。

更新日期:2019-11-18
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