Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture

An orthomorphism of a finite group $G$ is a bijection $\phi \text{\hspace{0.17em}⁣}:G\to G$ such that $g\mapsto g^{-1}\phi (g)$ is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when $G$ is abelian, for any $k\ge 2$ dividing $\mathrm{\mid }G\mathrm{\mid }-1$, there exists an orthomorphism of $G$ fixing the identity and permuting the remaining elements as products of disjoint $k$-cycles. We prove this conjecture for all sufficiently large groups.