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

Abstract

An orthomorphism of a finite group G is a bijection φ : G → G such that g → g −1 φ(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. We prove this conjecture for all sufficiently large groups.