Noether's problem for groups of order 243

Chu, Huah; Hoshi, Akinari; Hu, Shou-Jen; Kang, Ming-chang

doi:10.1016/j.jalgebra.2015.03.010

Cited by 13 publications

(6 citation statements)

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“…Similarly, Theorem 1.15 sheds some light on the understanding of the rationality problem of (G) for a group G of order p 5 where p = 5 or p = 7: If G belongs to the isoclinism family Φ i where i = 6, 7, 10, then (G) is not retract -rational. Applying the same arguments as in [CHHK,page 238, the first paragraph], it is easy to deduce that (G) is -rational if G is a group of order 5 5 or 7 5 and G belongs to the isoclinism family Φ i where 1 ≤ i ≤ 4 or 8 ≤ i ≤ 9.…”

Section: Introductionmentioning

confidence: 90%

Degree three unramified cohomology groups and Noether's problem for groups of order 243

2020

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Let k be a field and G be a finite group acting on the rational function field k(xg : g ∈ G) by k-automorphisms defined as h(xg) = x hg for any g, h ∈ G. We denote the fixed field k(xg : g ∈ G) G by k(G). Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is well-known that if (G) is stably rational over , then all the unramified cohomology groups H i nr ( (G), É/ ) = 0 for i ≥ 2. Hoshi, Kang and Kunyavskii [HKK] showed that, for a p-group of order p 5 (p: an odd prime number), H 2 nr ( (G), É/ ) = 0 if and only if G belongs to the isoclinism family Φ 10 . When p is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] exhibit some p-groups G which are of the form of a central extension of certain elementary abelian p-group by another one with H 2 nr ( (G), É/ ) = 0 and H 3 nr ( (G), É/ ) = 0. However, it is difficult to tell whether H 3 nr ( (G), É/ ) is non-trivial if G is an arbitrary finite group. In this paper, we are able to determine H 3 nr ( (G), É/ ) where G is any group of order p 5 with p = 3, 5, 7. Theorem 1. Let G be a group of order 3 5 . Then H 3 nr ( (G), É/ ) = 0 if and only if G belongs to the isoclinism family Φ 7 . Theorem 2. If G is a group of order 3 5 , then the fixed field (G) is rational if and only if G does not belong to the isoclinism families Φ 7 and Φ 10 . Theorem 3. Let G be a group of order 5 5 or 7 5 . Then H 3 nr ( (G), É/ ) = 0 if and only ifG belongs to the isoclinism families Φ 6 , Φ 7 or Φ 10 . Theorem 4. If G is the alternating group An, the Mathieu group M 11 , M 12 , the Janko group J 1 or the group P SL 2 ( q ), SL 2 ( q ), P GL 2 ( q ) (where q is a prime power), then H d nr ( (G), É/ ) = 0 for any d ≥ 2. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.

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Section: Introductionmentioning

confidence: 90%

Degree three unramified cohomology groups and Noether's problem for groups of order 243

2020

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“…In [HKK13] and [CHHK15], not only the evaluation of the Bogomolov multiplier B 0 (G) and the k-rationality of k(G) but also the k-isomorphisms between k(G 1 ) and k(G 2 ) for some groups G 1 and G 2 belonging to the same isoclinism family were given.…”

Section: Theorem 24 (Endo and Miyata [Em73 Theorem 23])mentioning

confidence: 99%

Noether's problem and rationality problem for multiplicative invariant fields: a survey

Hoshi¹

2020

Preprint

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In this paper, we give a brief survey of recent developments on Noether's problem and rationality problem for multiplicative invariant fields including author's recent papers Hoshi [Hos15] about Noether's problem over É, Hoshi, Kang and Kunyavskii [HKK13], Chu, Hoshi, Hu and Kang [CHHK15], Hoshi [Hos16] and Hoshi, Kang and Yamasaki [HKY16] about Noether's problem over , and Hoshi, Kang and Kitayama [HKK14] and Hoshi, Kang and Yamasaki [HKY] about rationality problem for multiplicative invariant fields. Contents § 1. Introduction § 2. Noether's problem over É § 3. Noether's problem over and unramified Brauer groups § 4. Rationality problem for multiplicative invariant fields

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“…The notion of Saltman's unramified Brauer group is extended to the higher degree unramified cohomology groups by Colliot-Thélène and Ojanguren [11]. For recent development of Noether's problem, we refer to works of A. Hoshi, M.-C. Kang, H. Kitayama, and A. Yamasaki and others [10,19,23,24,25,26,30,29,41,55]. Surely, the reader can also easily find more in the literature.…”

Section: Introductionmentioning

confidence: 99%

Chow’s Theorem for Semi-abelian Varieties and Bounds for Splitting Fields of Algebraic Tori

Yu¹

2019

Acta. Math. Sin.-English Ser.

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Noether's problem for groups of order 243

Cited by 13 publications

References 35 publications

Degree three unramified cohomology groups and Noether's problem for groups of order 243

Degree three unramified cohomology groups and Noether's problem for groups of order 243

Noether's problem and rationality problem for multiplicative invariant fields: a survey

Chow’s Theorem for Semi-abelian Varieties and Bounds for Splitting Fields of Algebraic Tori

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