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 $$<\alpha ,\beta ,\gamma \mid {\alpha }^{p^a}={\beta }^{p^b}={\gamma }^{p^c}=1,[\alpha ,\gamma ]=1,[\gamma ,\beta ]={\alpha }^{p^r},[\alpha ,\beta ]={\gamma }^{p^e}>,$$ where $n>a\ge b\ge c\ge 1$. 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 $k$.

This paper has been withdrawn by the author due to a crucial error in Corollary 2.2

References in corpus (2)

• Noether's problem and unramified Brauer groups
• Bogomolov multipliers and retract rationality for semi-direct products