论文标题
二次准群的同构
Isomorphisms of quadratic quasigroups
论文作者
论文摘要
Let $\mathbb{F}$ be a finite field of odd order and $a,b\in\mathbb{F}\setminus\{0,1\}$ be such that $χ(a) = χ(b)$ and $χ(1-a)=χ(1-b)$, where $χ$ is the extended quadratic character.令$ q_ {a，b} $为$ \ mathbb {f} $在$（x，y）\ mapsto x+a（y-x）$（y-x）\ ge 0 $ 0 $和$（x，x，x，x，y）\ mapsto x+b（y-x）$（y-x）$（y-x）$（y-x）$（y-x）$（y-x）$（y-x）$χ（y-χ（y-χ（y-百x）中），$（x，y）\ mapsto x+a（y-x）$定义为Quasigroup。我们证明$ q_ {a，b} \ cong q_ {c，d} $，并且仅当$ \ {a，b \} = \ {α（c），α（d）\} $ for in \ textrm {auttrm {aut}（aut}）（\ mathbb {f}）$。我们还表征了$ \ textrm {aut}（q_ {a，b}）$并展示进一步的属性，包括确定$ q_ {a，b} $是steiner quasigroup或交换性，熵，距离，左或右分配或右分配，灵活，灵活或半对称性。在证明我们的结果时，我们还表征了$ q_ {a，b} $的最小子量集。
Let $\mathbb{F}$ be a finite field of odd order and $a,b\in\mathbb{F}\setminus\{0,1\}$ be such that $χ(a) = χ(b)$ and $χ(1-a)=χ(1-b)$, where $χ$ is the extended quadratic character. Let $Q_{a,b}$ be the quasigroup upon $\mathbb{F}$ defined by $(x,y)\mapsto x+a(y-x)$ if $χ(y-x) \ge 0$, and $(x,y)\mapsto x+b(y-x)$ if $χ(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\}= \{α(c),α(d)\}$ for some $α\in \textrm{aut}(\mathbb{F})$. We also characterise $\textrm{aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results we also characterise the minimal subquasigroups of $Q_{a,b}$.
