Permutation polynomials of the form \(x^r f(x^{(q-1)/d)}\) and their group structure. (English) Zbl 0737.11040

A polynomial \(f(x)\) in \(\mathbb F_q[x]\) is called a permutation polynomial of the finite field \(\mathbb F_q\), if it induces a bijective map from \(\mathbb F_q\) to itself. The paper gives a systematic treatment of permutation polynomials over \(\mathbb F_q\) of the form \(x^rf(x^{(q-1)/d})\) and also determines their group structure. The group \(G(d,q)\) of permutation polynomials of this form is shown to be isomorphic to a generalized wreath product. The subgroup of \(G(d,q)\) consisting of all permutation polynomials of the form \(x^rf(x^{(q-1)/d})^d\), \(\deg(f)<d\), \((r,q-1) = 1\) and \(d\mid(q-1)\) is also considered.


11T06 Polynomials over finite fields
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