## Kyoto Journal of Mathematics

### Invariants of wreath products and subgroups of $S_{6}$

#### Abstract

Let $G$ be a subgroup of $S_{6}$, the symmetric group of degree 6. For any field $k$, $G$ acts naturally on the rational function field $k(x_{1},\ldots,x_{6})$ via $k$-automorphisms defined by $\sigma\cdot x_{i}=x_{\sigma(i)}$ for any $\sigma\in G$ and any $1\le i\le 6$. We prove the following theorem. The fixed field $k(x_{1},\ldots,x_{6})^{G}$ is rational (i.e., purely transcendental) over $k$, except possibly when $G$ is isomorphic to $\operatorname{PSL}_{2}(\mathbb{F}_{5})$, $\operatorname{PGL}_{2}(\mathbb{F}_{5})$, or $A_{6}$. When $G$ is isomorphic to $\operatorname{PSL}_{2}(\mathbb{F}_{5})$ or $\operatorname{PGL}_{2}(\mathbb{F}_{5})$, then $\mathbb{C}(x_{1},\ldots,x_{6})^{G}$ is $\mathbb{C}$-rational and $k(x_{1},\ldots,x_{6})^{G}$ is stably $k$-rational for any field $k$. The invariant theory of wreath products will be investigated also.

#### Article information

Source
Kyoto J. Math., Volume 55, Number 2 (2015), 257-279.

Dates
Revised: 10 February 2014
Accepted: 20 February 2014
First available in Project Euclid: 11 June 2015

https://projecteuclid.org/euclid.kjm/1433982755

Digital Object Identifier
doi:10.1215/21562261-2871749

Mathematical Reviews number (MathSciNet)
MR3356073

Zentralblatt MATH identifier
06457494

#### Citation

Kang, Ming-chang; Wang, Baoshan; Zhou, Jian. Invariants of wreath products and subgroups of $S_{6}$. Kyoto J. Math. 55 (2015), no. 2, 257--279. doi:10.1215/21562261-2871749. https://projecteuclid.org/euclid.kjm/1433982755

#### References

• [AHK] H. Ahmad, M. Hajja, and M.-C. Kang, Rationality of some projective linear actions, J. Algebra 228 (2000), 643–658.
• [CHK] H. Chu, S.-J. Hu, and M.-C. Kang, Noether’s problem for dihedral 2-groups, Comment. Math. Helv. 79 (2004), 147–159.
• [Di] J. Dixmier, “Sur les invariants du groupe symétrique dans certaines représentations, II” in Topics in Invariant Theory (Paris, 1989/1990), Lecture Notes in Math. 1478, Springer, Berlin, 1991, 1–34.
• [DM] J. D. Dixon and B. Mortimer, Permutation Groups, Grad. Texts in Math. 163, Springer, New York, 1996.
• [EM] S. Endô and T. Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 (1973), 7–26.
• [Fl] P. Fleischmann, The Noether bound in invariant theory of finite groups, Adv. Math. 156 (2000), 23–32.
• [Fo] J. Fogarty, On Noether’s bound for polynomial invariants of a finite group, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 5–7.
• [FH] W. Fulton and J. Harris, Representation Theory, Grad. Texts in Math. 129, Springer, Berlin, 1991.
• [GMS] S. Garibaldi, A. Merkurjev, and J.-P. Serre, Cohomological Invariants in Galois Cohomology, Univ. Lecture Ser. 28, Amer. Math. Soc., Providence, 2003.
• [Ga] W. Gaschütz, Fixkörper von $p$-Automorphismengruppen reintranszendener Körpererweiterungen von $p$-Charakteristik, Math. Z. 71 (1959), 466–468.
• [Ha] M. Hajja, Rationality of finite groups of monomial automorphisms of $k(x,y)$, J. Algebra 109 (1987), 46–51.
• [HK1] M. Hajja and M.-C. Kang, Finite group actions on rational function fields, J. Algebra 149 (1992), 139–154.
• [HK2] M. Hajja and M.-C. Kang, Three-dimensional purely monomial group actions, J. Algebra 170 (1994), 805–860.
• [HK3] M. Hajja and M.-C. Kang, Some actions of symmetric groups, J. Algebra 177 (1995), 511–535.
• [HHR] K.-I. Hashimoto, A. Hoshi, and Y. Rikuna, Noether’s problem and $\mathbb{Q}$-generic polynomials for the normalizer of the $8$-cycle in $S_{8}$ and its subgroups, Math. Comp. 77 (2008), 1153–1183.
• [HT1] K.-I. Hashimoto and H. Tsunogai, Generic polynomials over $\mathbf{Q}$ with two parameters for the transitive permutation groups of degree five, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), 142–145.
• [HT2] K.-I. Hashimoto and H. Tsunogai, “Noether’s problem for transitive permutation groups of degree $6$” in Galois-Teichmuller Theory and Arithmetic Geometry, Adv. Stud. Pure Math. 63, Math. Soc. Japan, Tokyo, 2012, 189–220.
• [Ho] A. Hoshi, Multiplicative quadratic forms on algebraic varieties and Noether’s problem for meta-abelian groups, Ph.D. dissertation, Waseda University, Tokyo, 2005, available at http://dspace.wul.waseda.ac.jp/dspace/handle/2065/3004 (accessed 11, March 2015).
• [HR] A. Hoshi and Y. Rikuna, Rationality problem of three-dimensional purely monomial group actions: The last case, Math. Comp. 77 (2008), 1823–1829.
• [Is] I. M. Isaacs, Finite Group Theory, Grad. Stud. Math. 92, Amer. Math. Soc., Providence, 2008.
• [Ka1] M.-C. Kang, Rationality problem of $\operatorname{GL}_{4}$ group actions, Adv. Math. 181 (2004), 321–352.
• [Ka2] M.-C. Kang, Noether’s problem for dihedral 2-groups, II, Pacific J. Math. 222 (2005), 301–316.
• [KP] M.-C. Kang and B. Plans, Reduction theorems for Noether’s problem, Proc. Amer. Math. Soc. 137 (2009), 1867–1874.
• [KW] M.-C. Kang and B. Wang, Rational invariants for subgroups of $S_{5}$ and $S_{7}$, J. Algebra 413 (2014), 345–363.
• [Ku] H. Kuniyoshi, On a problem of Chevalley, Nagoya Math. J. 8 (1955), 65–67.
• [Kuy1] W. Kuyk, On the inversion problem of Galois theory (in Dutch), Ph.D. dissertation, Vrije Universiteit te Amsterdam, Amsterdam, 1960.
• [Kuy2] W. Kuyk, On a theorem of E. Noether, Indag. Math. 26 (1964), 32–39.
• [Le] H. W. Lenstra Jr., Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325.
• [Ma] T. Maeda, Noether’s problem for $A_{5}$, J. Algebra 125 (1989), 418–430.
• [No] E. Noether, Rationale Funktionenkörper, Jber. Deutsch. Math.-Verein. 22 (1913), 316–319.
• [Ro] D. J. S. Robinson, A course in the theory of groups, Grad. Texts in Math. 80, Springer, Berlin, 1982.
• [Sa] D. J. Saltman, Generic Galois extensions and problems in field theory, Adv. Math. 43 (1982), 250–283.
• [Sh] N. I. Shepherd-Barron, Invariant theory for $S_{5}$ and the rationality of $M_{6}$, Compos. Math. 70 (1989), 13–25.
• [Sm] L. Smith, Polynomial Invariants of Finite Groups, Res. Notes Math. 6, Peters, Wellesley, Mass., 1995.
• [Sw] R. G. Swan, “Noether’s problem in Galois theory” in Emmy Noether in Bryn Mawr, Springer, Berlin, 1983, 21–40.
• [Ts] H. Tsunogai, Noether’s problem for Sylow subgroups of symmetric groups and its application (in Japanese), Sūrikaisekikenkyūsho Kūkyūroku 1451 (2005), 265–274.
• [WZ] B. Wang and J. Zhou, Rationality problem for transitive subgroups of $S_{8}$, preprint, arXiv:1402.1675v1 [math.AG].
• [Za] O. Zariski, On Castelnuovo’s criterion of rationality $p_{a}=P_{2}=0$ of an algebraic surface, Illinois J. Math. 2 (1958), 303–315.
• [Zh] J. Zhou, Rationality for subgroups of $S_{6}$, to appear in Comm. Algebra, preprint, arXiv:1308.0409v2 [math.AG].