Isomorphisms of quadratic quasigroups

TL;DR

This paper classifies isomorphisms of certain quadratic quasigroups over finite fields, characterizes their automorphism groups, and explores properties like commutativity, distributivity, and subquasigroup structure.

Contribution

It provides a complete characterization of when two such quadratic quasigroups are isomorphic and describes their automorphism groups and key algebraic properties.

Findings

Isomorphism of quadratic quasigroups determined by field automorphisms.

Automorphism groups of these quasigroups characterized.

Conditions for properties like commutativity and distributivity established.

Abstract

Let $\mathbb{F}$ be a finite field of odd order and $a,b\in\mathbb{F}\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where $\chi$ 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 $\chi(y-x) \ge 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\}= \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \textrm{aut}(\mathbb{F})$. 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 $Q_{a,b}$.