Noether's problem for p-groups with three generators

Abstract

Let $p$ be an odd prime and $G$ be a nonabelian group of order $p^{n}$ with the presentation $$<α,β,γ\mid α^{p^{a}}=β^{p^{b}}=γ^{p^{c}}=1, [α,γ]=1,[γ,β]=α^{p^{r}},[α,β]=γ^{p^{e}}>,$$ where $n>a\geq b\geq c\geq 1$. Let $k$ be a field containing a primitive $p^{a}$-th root of unity and $G$ act on the rational function field $k(x_{h}:h\in G)$ by $g\cdot x_{h}=x_{gh}$ for all $g,h\in G$. In this note, we prove that the fixed field $k(G)=k(x_{h}:h\in G)^{G}$ is rational over $k$. As a corollary, we prove that if $k$ contains a primitive $p^{4}$-th root of unity and $G$ is a nonabelian group of order $p^{5}$ generated by three elements, then $k(G)$ is rational over $k$.