Isomorphisms of quadratic quasigroups

Abstract

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 $Q_{a,b}$.