Abstrakt
Abstract Let F be a finite field of odd order and a, b ELEMENT OF F \ {0, 1} be such that χ(a) = χ(b) and χ(1 - a) = χ(1 - b), where χ is the extended quadratic character on F. Let Qa,b be the quasigroup over F defined by (x, y) 7RIGHTWARDS ARROW x + a(y - x) if χ(y - x) > 0, and (x, y) 7RIGHTWARDS ARROW x + b(y - x) if χ(y - x) = -1.
We show that Qa,b TILDE OPERATOR+D91= Qc,d if and only if {a, b} = {α(c), α(d)} for some α ELEMENT OF Aut(F). We also characterize Aut(Qa,b) and exhibit further properties, including establishing when Qa,b is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric.
In proving our results, we also characterize the minimal subquasigroups of Qa,b.