Invariants of wreath products and subgroups of S_6

Abstract

Let $G$ be a subgroup of $S_6$, the symmetric group of degree 6. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,...,x_6)$ via $k$-automorphisms defined by $\sigma \cdot x_i=x_{\sigma (i)}$ for any $\sigma \in G$, any $1\le i\le 6$. Theorem. The fixed field $k(x_1,...,x_6{)}^G$ is rational (=purely transcendental) over $k$, except possibly when $G$ is isomorphic to $PS{L}_2({\mathbf{F}}_5)$, $PG{L}_2({\mathbf{F}}_5)$ or $A_6$. When $G$ is isomorphic to $PS{L}_2({\mathbf{F}}_5)$ or $PG{L}_2({\mathbf{F}}_5)$, then $\mathbf{C}(x_1,...,x_6{)}^G$ is $\mathbf{C}$-rational and $k(x_1,...,x_6{)}^G$ is stably $k$-rational for any field $k$. The invariant theory of wreath products will be investigated also.

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