Noether's Problem and the Unramified Brauer Group for Groups of Order 64

Chu, Huah; Hu, Shou-Jen; Kang, Ming-chang; Kunyavskiı̆, Boris

doi:10.1093/imrn/rnp217

Cited by 26 publications

(31 citation statements)

References 22 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%

“…With groups of order 64, there are 237 groups with the same property out of a total of 267 groups. Again, this may be compared with known results [5]. Finally, one may use Theorem 1.1 on the Sylow subgroups of a given group rather than using Corollary 1.2 directly, thus potentially obtaining a better bound on commuting probability that ensures triviality of the Bogomolov multiplier.…”

Section: Applicationsmentioning

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%

Section: B 0 -Minimal Groupsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

See 2 more Smart Citations

Universal commutator relations, Bogomolov multipliers, and commuting probability

Jezernik¹,

Moravec²

2015

Journal of Algebra

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“…Moreover Chu, Hu, Kang and Kunyavskii [CHKK10] investigated the case where G is a group of order 64 as follows. There exist exactly 267 non-isomorphic groups of order 64.…”

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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The Bogomolov multiplier of rigid finite groups

Kang

Kunyavskiı̆

2014

Arch. Math.

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The Bogomolov multiplier of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. This invariant of G plays an important role in birational geometry of quotient spaces V /G. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of G in the sense that it has no outer class-preserving automorphisms.

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Noether's Problem and the Unramified Brauer Group for Groups of Order 64

Cited by 26 publications

References 22 publications

Universal commutator relations, Bogomolov multipliers, and commuting probability

Universal commutator relations, Bogomolov multipliers, and commuting probability

Rationality Problem for Algebraic Tori

The Bogomolov multiplier of rigid finite groups

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