Kyoto Journal of Mathematics

Invariants of wreath products and subgroups of S 6

Ming-chang Kang, Baoshan Wang, and Jian Zhou

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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 , , x 6 ) via k -automorphisms defined by σ x i = x σ ( i ) for any σ G and any 1 i 6 . We prove the following theorem. The fixed field k ( x 1 , , x 6 ) G is rational (i.e., purely transcendental) over k , except possibly when G is isomorphic to PSL 2 ( F 5 ) , PGL 2 ( F 5 ) , or A 6 . When G is isomorphic to PSL 2 ( F 5 ) or PGL 2 ( F 5 ) , then C ( x 1 , , x 6 ) G is C -rational and k ( x 1 , , x 6 ) G is stably k -rational for any field k . The invariant theory of wreath products will be investigated also.

Article information

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

Received: 18 September 2013
Revised: 10 February 2014
Accepted: 20 February 2014
First available in Project Euclid: 11 June 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]
Secondary: 14E08: Rationality questions [See also 14M20]

Noether’s problem rationality problem wreath products


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.

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