Degree three unramified cohomology groups and Noether's problem for groups of order $243$

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 $C(G)$ is stably rational over $C$, then all the unramified cohomology groups H_[nr}^i(C(G),Q/Z)=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_[nr}^2(C(G),Q/Z)\neq 0 if and only if $G$ belongs to the isoclinism family ${\mathrm{\Phi }}_{10}$. When $p$ is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] 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_[nr}^2(C(G),Q/Z)=0 and H_[nr}^3(C(G),Q/Z)\neq 0. However, it is difficult to tell whether H_[nr}^3(C(G),Q/Z) is non-trivial if $G$ is an arbitrary finite group. In this paper, we are able to determine H_[nr}^3(C(G),Q/Z) where $G$ is any group of order $p^5$ with $p=3,5,7$. Theorem 1. Let $G$ be a group of order $3^5$. Then H_[nr}^3(C(G),Q/Z)\neq 0 if and only if $G$ belongs to ${\mathrm{\Phi }}_7$. Theorem 2. If $G$ is a group of order $3^5$, then the fixed field $C(G)$ is rational if and only if $G$ does not belong to ${\mathrm{\Phi }}_7$ and ${\mathrm{\Phi }}_{10}$. Theorem 3. Let $G$ be a group of order $5^5$ or $7^5$. Then H_[nr}^3(C(G),Q/Z)\neq 0 if and only if $G$ belongs to ${\mathrm{\Phi }}_6$, ${\mathrm{\Phi }}_7$ or ${\mathrm{\Phi }}_{10}$. Theorem 4. If $G$ is the alternating group $A_n$, the Mathieu group $M_{11}$, $M_{12}$, the Janko group $J_1$ or the group $PS{L}_2(F_q)$, $S{L}_2(F_q)$, $PG{L}_2(F_q)$ (where $q$ is a prime power), then H_[nr}^d(C(G),Q/Z)=0 for any $d\ge 2$. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.