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Complementary permutations for abelian groups

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LetG be an additively written abelian group and leth: G → G be a given function. M. Hall Jr. (1952) and L. Fuchs (1958) already answered the following question. For what functionsh: G → G does the functional equationσ(x) + τ(x) = h(x) (x ∈ G) have as its solution a pair of permutationsσ andτ ofG? In this paper, we give explicit constructions of such a pairσ, τ in a number of cases, in particular whenh(x) ≡ x andG is finite. We further determine the finite groupsG where the latterσ, τ can be chosen to be automorphisms.

In the case whereG is an infinite topological group, we study in how farσ andτ can be chosen as Borel measurable permutations, given thath: G → G itself is Borel measurable.

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Kemperman, J.H.B., Ott, T.J. Complementary permutations for abelian groups. Aeq. Math. 48, 262–282 (1994). https://doi.org/10.1007/BF01832989

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AMS (1980) subject classification

  • Primary 39B50
  • Secondary 20Kxx, 68R05