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For any field $k$, $G$ acts naturally on the rational function field $k(x_1,x_2,\\ldots,x_n)$ via $k$-automorphisms defined by $\\sigma\\cdot x_i=x_{\\sigma(i)}$ for any $\\sigma\\in G$, any $1\\le i\\le n$. Theorem. If $n\\le 5$, then the fixed field $k(x_1,\\ldots,x_n)^G$ is purely transcendental over $k$. 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