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A strong complete mapping of a group G is a bijection θ : G → G for which both mappings x → xθ(x) and x → x −1 θ(x) are bijections. We characterize finite abelian groups that admit strong complete mappings, thus solving a problem posed by Horton in 1990. We also prove the existence of strong complete mappings for countably infinite groups.fatcat:4zhxlvpcdjeulehyqug7jniqni