Let Ω = H1⊕…⊕Hn be an abelian group of permutations of a finite non-empty set S. If Hi is generated by φi, let sφi(α) denote the length of the cycle of φi containing α. For any function f on S, let A(f,Ω) = (φ ∈ Ω|fφ = f). In Theorem 2 we show that, if for every i ≠ j and α ∈ S, Sφi(α) and Sφj(α) are relatively prime, then A(f, Ω) = A(f, H1)⊕…⊕A(f, Hn) for all f, while in Theorem 3 we prove the natural converse.