Quasigroup automorphisms and the Norton-Stein complex

Abstract

Suppose that $d>1$ is the largest power of two that divides the order of a finite quasigroup $Q$. It then follows that each automorphism of $Q$ must contain a cycle of length not divisible by $d$ in its disjoint cycle decomposition. The proof is obtained by considering the action induced by the automorphism on a certain orientable surface originally described in a more restricted context by Norton and Stein.