Bruck–Murdoch–Toyoda theorem

Statement

The Bruck–Murdoch–Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group $A$ and two commuting automorphisms $\varphi$ and $\psi$ of $A$, define an operation $\bullet$ on $A$ by

$$x \bullet y = \varphi(x) + \psi(y) + c,$$

where $c$ some fixed element of $A$. It is not hard to prove that $A$ forms a medial quasigroup under this operation. The Bruck–Murdoch-Toyoda theorem states that every medial quasigroup is of this form, i.e. is isomorphic to a quasigroup defined from an abelian group in this way. In particular, every medial quasigroup is isotopic to an abelian group.

The result was obtained independently in 1941 by Murdoch and Toyoda. It was then rediscovered by Bruck in 1944.