Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is well-known that if $\bC(G)$ is stably rational over $\bC$, then all the unramified cohomology groups $H_{\rm nr}^i(\bC(G),\bQ/\bZ)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$-group of order $p^5$ ($p$: an odd prime number), $H_{\rm nr}^2(\bC(G),\bQ/\bZ)\neq 0$ if and only if $G$ belongs to the isoclinism family $\Phi_{10}$. When $p$ is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY] exhibit some $p$-groups $G$ which are of the form of a central extension of certain elementary abelian $p$-group by another one with $H_{\rm nr}^2(\bC(G),\bQ/\bZ)= 0$ and $H_{\rm nr}^3(\bC(G),\bQ/\bZ) \neq 0$. However, it is difficult to tell whe 查看全文>>