## 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.

### MSC:

 11T06 Polynomials over finite fields
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### References:

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