Cycle type in Hall–Paige: a proof of the Friedlander–Gordon–Tannenbaum conjecture

An orthomorphism of a finite group G is a bijection φ G→ G such that g↦ g⁻¹φ (g) is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when G is abelian, for any k≥ 2 dividing $|G|-1$ , there exists an orthomorphism of G fixing the identity and permuting the remaining elements as products of disjoint k -cycles as long as the Sylow $2$ -subgroups of G are trivial or noncyclic. We prove this conjecture for all sufficiently large groups.