Rational invariants for subgroups of S_5 and S_7

Abstract

Let $G$ be a subgroup of $S_n$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,x_2,\dots ,x_n)$ 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 $k(x_1,\dots ,x_n{)}^G$ is purely transcendental over $k$. We will show that $\mathbf{C}(x_1,\dots ,x_7{)}^G$ is also purely transcendental over $\mathbf{C}$ if $G$ is any transitive subgroups of $S_7$ other than $A_7$; a similar result is valid for solvable transitive subgroups of $S_{11}$.

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