In this paper we discuss piece-wise permutation polynomials (PP). The method is by combining the AGW lemma and module structure, and we get a generalized five lemma. Then we apply the results to additive and multiplicative structures of finite fields. This gives a unified treatment and a framework of extensive existing PPs in the literature. More precisely, we deal with PPs of the following forms [Multiplicative Structure] xau(xqn-1d)\documentclass[12pt]{minimal}
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\begin{document}$$x^au(x^{\frac{q^n-1}{d}})$$\end{document}, where a∈N\documentclass[12pt]{minimal}
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\begin{document}$$a\in \mathbb {N}$$\end{document}, d|qn-1\documentclass[12pt]{minimal}
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\begin{document}$$d|q^n-1$$\end{document}, u(x)∈Fqn[x]\documentclass[12pt]{minimal}
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\begin{document}$$u(x)\in \mathbb {F}_{q^n}[x]$$\end{document}.[Additive Structure] L1+u(L2+δ)\documentclass[12pt]{minimal}
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\begin{document}$$L_{1}+u(L_{2}+\delta )$$\end{document}, where L1\documentclass[12pt]{minimal}
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\begin{document}$$L_1$$\end{document}, L2\documentclass[12pt]{minimal}
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\begin{document}$$L_2$$\end{document} are linearized polynomials, u(x)∈Fqn[x]\documentclass[12pt]{minimal}
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\begin{document}$$u(x)\in \mathbb {F}_{q^n}[x]$$\end{document}, δ∈Fqn\documentclass[12pt]{minimal}
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\begin{document}$$\delta \in \mathbb {F}_{q^n}$$\end{document}. If L2\documentclass[12pt]{minimal}
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\begin{document}$$L_{2}$$\end{document} is the trace function Tr(x)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Tr}(x)$$\end{document}, this concerns both additive and multiplicative structures.
