Noether's problems for groups of order 243

Abstract

Let $k$ be any field, $G$ be a finite group. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $h\cdot x_g=x_{hg}$ for any $g,h\in G$. Denote by $k(G)=k(x_g:g\in G)^G$ the fixed field. Noether's problem asks, under what situations, the fixed field $k(G)$ will be rational (= purely transcendental) over $k$. According to the data base of GAP there are $10$ isoclinism families for groups of order $243$. It is known that there are precisely $3$ groups $G$ of order $243$ (they consist of the isoclinism family $\Phi_{10}$) such that the unramified Brauer group of $\bm{C}(G)$ over $\bm{C}$ is non-trivial. Thus $\bm{C}(G)$ is not rational over $\bm{C}$. We will prove that, if $\zeta_9 \in k$, then $k(G)$ is rational over $k$ for groups of order $243$ other than these $3$ groups, except possibly for groups belonging to the isoclinism family $\Phi_7$.