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 $k(x_1,\dots ,x_6)$ via $k$-automorphisms defined by $\sigma \cdot x_i=x_{\sigma \left(i\right)}$ for any $\sigma \in G$ and any $1\le i\le 6$. We prove the following theorem. The fixed field $k(x_1,\dots ,x_6{)}^G$ is rational (i.e., purely transcendental) over $k$, except possibly when $G$ is isomorphic to ${PSL}_2\left({\mathbb{F}}_5\right)$, ${PGL}_2\left({\mathbb{F}}_5\right)$, or $A_6$. When $G$ is isomorphic to ${PSL}_2\left({\mathbb{F}}_5\right)$ or ${PGL}_2\left({\mathbb{F}}_5\right)$, then $\mathbb{C}(x_1,\dots ,x_6{)}^G$ is $\mathbb{C}$-rational and $k(x_1,\dots ,x_6{)}^G$ is stably $k$-rational for any field $k$. The invariant theory of wreath products will be investigated also.