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Automorphisms of functions in abelian permutation groups

Published online by Cambridge University Press:  18 May 2009

Stephen D. Cohen  and
Gary L. Mullen
Stephen D. Cohen
Affiliation:
University of Glasgow, Glasgow G12 8QW
Gary L. Mullen
Affiliation:
The Pennsylvania State University, Shenango Valley Campus, 147 Shenango Avenue, Sharon, Pennsylvania 16146, U.S.A.
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 16 , Issue 2 , September 1975 , pp. 107 - 108
Copyright
Copyright © Glasgow Mathematical Journal Trust 1975

References

REFERENCE

1.Carlitz, L., Invariantive theory of equations in a finite field, Trans. Amer. Math. Soc. 75 (1953), 405–427.CrossRefGoogle Scholar