Groups of permutations generated by function-linear translator pairs

In [8], G. Kyureghyan showed that the function F(x)=x+γf(x) is a permutation of Fqm when f:Fqm→Fq is a function, γ∈Fqm is a b-linear translator for f for some b(≠−1)∈Fq. His idea has been extended in [19] by Qin et al. and in [9] by M. Kyureghyan and Abrahamyan to finitely many function-linear translator pairs. In this paper, we study the permutations generated by function-linear translator pairs along G. Kyureghyan's idea and prove that these permutations form groups whose group structures are well understood.