Noether's problem for groups of order 32

Chu, Huah; Hu, Shou-Jen; Kang, Ming-chang; Prokhorov, Yuri

doi:10.1016/j.jalgebra.2008.07.007

Cited by 43 publications

(16 citation statements)

References 18 publications

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“…It follows from [5] that B 0 (G) ∼ = Z/2Z. Application of the algorithm developed in [24] shows that B 0 (G) is generated by the element (g 3 g 2 )(g 4 g 1 ) in G G. The group G is one of the groups of the smallest order that have a nontrivial Bogomolov multiplier [6,5], so it is also of minimal order amongst all B 0 -minimal groups.…”

Section: B 0 -Minimal Groupsmentioning

confidence: 99%

“…Making use of recent results on Bogomolov multipliers of p-groups of small orders [13,14,5,6,4,18], we determine the B 0 -minimal families of rank at most 6 for odd primes p, and those of rank at most 7 for p = 2. In stating the proposition, the classifications [16,17] are used.…”

Section: B 0 -Minimal Groupsmentioning

confidence: 99%

“…Now let p = 2. It is shown in [6,5] that the groups of minimal order having nontrivial Bogomolov multipliers are exactly the groups forming the stem of the isoclinism family Γ 16 of [12], so this family is B 0 -minimal. In the notation of [17], it corresponds to Φ 16 .…”

Section: B 0 -Minimal Groupsmentioning

confidence: 99%

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Universal commutator relations, Bogomolov multipliers, and commuting probability

Jezernik¹,

Moravec²

2015

Journal of Algebra

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Let G be a finite p-group. We prove that whenever the commuting probability of G is greater than (2p 2 +p −2)/p 5 , the unramified Brauer group of the field of G-invariant functions is trivial. Equivalently, all relations between commutators in G are consequences of some universal ones. The bound is best possible, and gives a global lower bound of 1/4 for all finite groups. The result is attained by describing the structure of groups whose Bogomolov multipliers are nontrivial, and Bogomolov multipliers of all of their proper subgroups and quotients are trivial. Applications include a classification of p-groups of minimal order that have nontrivial Bogomolov multipliers and are of nilpotency class 2, a nonprobabilistic criterion for the vanishing of the Bogomolov multiplier, and establishing a sequence of Bogomolov's absolute γ-minimal factors which are 2-groups of arbitrarily large nilpotency class, thus providing counterexamples to some of Bogomolov's claims. In relation to this, we fill a gap in the proof of triviality of Bogomolov multipliers of finite almost simple groups.

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Section: B 0 -Minimal Groupsmentioning

confidence: 99%

Section: B 0 -Minimal Groupsmentioning

confidence: 99%

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Universal commutator relations, Bogomolov multipliers, and commuting probability

Jezernik¹,

Moravec²

2015

Journal of Algebra

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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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“…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: 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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Noether's problem for groups of order 32

Abstract: field K (G) will be rational (= purely transcendental) over K .Theorem. Let G be a finite group of order 32 with exponent e. If char K = 2 or K is any field containing a primitive eth root of unity, then K (G) is rational over K .

Cited by 43 publications

References 18 publications

Universal commutator relations, Bogomolov multipliers, and commuting probability

Universal commutator relations, Bogomolov multipliers, and commuting probability

Rationality Problem for Algebraic Tori

Noether's problem for groups of order 243

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