Isomorphisms of quadratic quasigroups
Let F be a finite field of odd order and
a,b in F ∖ {0,1} be such that χ(a) = χ(b)
and χ(1-a)=χ(1-b), where χ is the quadratic
character.
Let Qa,b be the quasigroup upon F defined by
(x,y) → x+a(y-x) if χ(y-x) ≥ 0, and (x,y)
→ x+b(y-x) if χ(y-x) = -1.
We show that Qa,b is isomorphic to Qc,d
if and only if {a,b} = {α(c),α(d)} for some
α in aut(F). We also characterise
aut(Qa,b) and exhibit further properties,
including establishing when Qa,b
is a Steiner quasigroup or is commutative, medial, left or right distributive,
flexible or semisymmetric.