## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 189 - 220

### Noether's problem for transitive permutation groups of degree 6

Kiichiro Hashimoto and Hiroshi Tsunogai

#### Abstract

Suppose that a finite group $G$ is realized in the Cremona group $\mathrm{Cr}_m (k)$, the group of $k$-automorphisms of the rational function field $K$ of $m$ variables over a constant field $k$. The most general version of Noether's problem is then to ask, whether the subfield $K^G$ consisting of $G$-invariant elements is again rational or not. This paper treats Noether's problem for various subgroups $G$ of $\mathfrak{S}_6$, the symmetric group of degree 6, acting on the function field $\boldsymbol{Q}(s, t, z)$ over $k = \boldsymbol{Q}$ of the moduli space $\mathcal{M}_{0,6}$ of $\mathbb{P}^1$ with ordered six marked points. We shall show that this version of Noether's problem has an affirmative answer for all but two conjugacy classes of transitive subgroups $G$ of $\mathfrak{S}_6$, by exhibiting explicitly a system of generators of the fixed field $\boldsymbol{Q}(s,t,z)^G$. In the exceptional cases $G \cong \mathfrak{A}_6, \mathfrak{A}_5$, the problem remains open.

#### Article information

**Dates**

Received: 23 May 2011

Revised: 4 April 2012

First available in Project Euclid:
24 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1540417819

**Digital Object Identifier**

doi:10.2969/aspm/06310189

**Mathematical Reviews number (MathSciNet)**

MR3051244

**Zentralblatt MATH identifier**

1325.12008

**Subjects**

Primary: 12F12: Inverse Galois theory

Secondary: 12F10: Separable extensions, Galois theory 14H05: Algebraic functions; function fields [See also 11R58] 14H10: Families, moduli (algebraic) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX] 20B35: Subgroups of symmetric groups

**Keywords**

Galois theory invariant theory permutation groups Cremona group cross ratios hyperelliptic curves

#### Citation

Hashimoto, Kiichiro; Tsunogai, Hiroshi. Noether's problem for transitive permutation groups of degree 6. Galois–Teichmüller Theory and Arithmetic Geometry, 189--220, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310189. https://projecteuclid.org/euclid.aspm/1540417819