Geodesic
Automorphisms of functions in abelian permutation groups
Glasgow mathematical journal, Tome 16 (1975) no. 2, pp. 107-108

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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. 
Cohen, Stephen D.; Mullen, Gary L. Automorphisms of functions in abelian permutation groups. Glasgow mathematical journal, Tome 16 (1975) no. 2, pp. 107-108. doi: 10.1017/S0017089500002597
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[1] 1.Carlitz, L., Invariantive theory of equations in a finite field, Trans. Amer. Math. Soc. 75 (1953), 405–427. Google Scholar | DOI

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