Rationality of P-Group Actions

Chu, Huah; Kang, Ming-chang

doi:10.1006/jabr.2000.8615

Cited by 49 publications

(34 citation statements)

References 14 publications

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“…In the case where p = 2, by Chu and Kang [CK01] and Chu, Hu, Kang and Prokhorov [CHKP08], if G is a group of order ≤ 32, then F (G) is F -rational. Moreover Chu, Hu, Kang and Kunyavskii [CHKK10] investigated the case where G is a group of order 64 as follows.…”

Section: Application Of Theorem 123mentioning

confidence: 99%

Rationality Problem for Algebraic Tori

Hoshi

Yamasaki

2017

Memoirs of the AMS

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Abstract. We give the complete stably rational classification of algebraic tori of dimensions 4 and 5 over a field k. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank 4 and 5 is given. We show that there exist exactly 487 (resp. 7, resp. 216) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 4, and there exist exactly 3051 (resp. 25, resp. 3003) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 5. We make a procedure to compute a flabby resolution of a G-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a G-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby G-lattices of rank up to 6 and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for G-lattices holds when the rank ≤ 4, and fails when the rank is 5. Indeed, there exist exactly 11 (resp. 131) G-lattices of rank 5 (resp. 6) which are decomposable into two different ranks. Moreover, when the rank is 6, there exist exactly 18 G-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that H 1 (G, F ) = 0 for any Bravais group G of dimension n ≤ 6 where F is the flabby class of the corresponding G-lattice of rank n. In particular, H 1 (G, F ) = 0 for any maximal finite subgroup G ≤ GL(n, ) where n ≤ 6. As an application of the methods developed, some examples of not retract (stably) rational fields over k are given.

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Section: Application Of Theorem 123mentioning

confidence: 99%

Rationality Problem for Algebraic Tori

Hoshi

Yamasaki

2017

Memoirs of the AMS

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“…In studying the rationality problem of these groups, new technical difficulties (different from the situations in [9,8,7]) arise. Fortunately we are able to discover new methods to solve these difficulties (see Step 4 of Case 1 in Section 4, Steps 3 and 5 of Case 1 in Section 5).…”

Section: Introductionmentioning

confidence: 98%

“…Hence we may focus on non-abelian groups. [9,Theorem 1.6].) Let G be a p-group of order p 3 or p 4 .…”

Section: Introductionmentioning

confidence: 99%

Noether's problem for groups of order 243

Chu

Hoshi

et al. 2015

Journal of Algebra

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will be rational (= purely transcendental) over k. According to the data base of GAP there are 10 isoclinism families for groups of order 243. It is known that there are precisely 3 groups G of order 243 (they consist of the isoclinism family Φ 10 ) such that the unramified Brauer group of C(G) over C is non-trivial. Thus C(G) is not rational over C. We will prove that, if ζ 9 ∈ k, then k(G) is rational over k for groups of order 243 other than these 3 groups, except possibly for groups belonging to the isoclinism family Φ 7 .

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“…But our knowledge about Noether's problem for nonabelian groups is rather incomplete. It is known that K (G) is rational if G is a transitive solvable subgroup of the symmetric group S p when p = 3, 5, 7, 11 [Furtwängler 1925], the quaternion group of order 8 [Grbner 1934], the alternating group A 5 [Maeda 1989;Kervaire and Vust 1989], P S L 2 (7), P Sp 4 (3) (such that the base fields K contain suitable quadratic fields of ‫)ޑ‬ [Kemper 1996] or finite reflection groups [Kemper and Malle 1999]. Noether's problem for metaabelian groups and dihedral groups is discussed in [Haeuslein 1971;Hajja 1983;Kang 2004].…”

Section: Introductionmentioning

confidence: 99%

“…(See [Bogomolov 1987] for pgroups of smaller orders.) If K is a field containing enough roots of unity, then K (G) is rational for any nonabelian group of order p 3 or p 4 [Chu and Kang 2001]. A result of Serre [1995, 3.5] (see also [Garibaldi et al 2003, Theorem 33.26 and Example 33.27, pp.…”

Section: Introductionmentioning

confidence: 99%

Noether’s problem for dihedral 2-groups II

Kang¹

2005

Pacific J. Math.

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is rational (= purely transcendental) over K . A result of Serre shows that ‫(ޑ‬G) is not rational when G is the generalized quaternion group of order 16. We shall prove that K (G) is rational over K if G is any nonabelian group of order 16 except when G is the generalized quaternion group of order 16. When G is the generalized quaternion group of order 16 and K (ζ 8 ) is a cyclic extension of K , then K (G) is also rational over K .

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Rationality of P-Group Actions

Abstract: Let p be a prime number, G a p-group of order ≤ p 4 K a field with char K = p. If the exponent of G is p e and K contains a primitive p e th root of unity, then K V G is rational (=purely transcendental) over K where ρ G → GL V is any linear representation of G over K.

Cited by 49 publications

References 14 publications

Rationality Problem for Algebraic Tori

Rationality Problem for Algebraic Tori

Noether's problem for groups of order 243

Noether’s problem for dihedral 2-groups II

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