Noether's problem for $p$-groups with an abelian subgroup of index $p$

Details Noether's problem for $p$-groups with an abelian subgroup of index $p$ Noether's problem for $p$-groups with an abelian subgroup of index $p$

2012-01-26

arxiv.org/abs/1201.5555v3

Mathematics

Algebraic Geometry

In version [v2] I corrected the statement of the main result. Also, I added a lemma describing the properties of the groups un

Scientific paper

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g...t x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational over $K$. {\bf Theorem.} Let $G$ be a group of order $p^n$ for $n\geq 2$ with an abelian subgroup $H$ of order $p^{n-1}$, and let $G$ be of exponent $p^e$. Assume that $H=H_1\times H_2\times...\times H_s$ for some $s\geq 1$ where $H_j\simeq C_{p^{i_j}}\times (C_p)^{k_j}$ and $H_j$ is normal in $G$ for $1\leq j\leq s, 0\leq k_j,1\leq i_1\leq i_2\leq...\leq i_s$. Assume also that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$.

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