Abstract
The Bogomolov multiplier B 0(G) 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. The triviality of the Bogomolov multiplier is an obstruction to Noether’s problem. We show that if G is a central product of G 1 and G 2, regarding K i ≤ Z(G i ), i = 1, 2, and θ: G 1 → G 2 is a group homomorphism such that its restriction \(\theta {|_{{K_1}}}:{K_1} \to {K_2}\) is an isomorphism, then the triviality of B 0(G 1/K 1),B 0(G 1) and B 0(G 2) implies the triviality of B 0(G). We give a positive answer to Noether’s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement for the roots of unity).
Similar content being viewed by others
References (wenxian biaoti)
Ahmad, H., Hajja, S., Kang, M.: Rationality of some projective linear actions. J. Algebra, 228, 643–658 (2000)
Ahmad, A., Magidin, A., Morse, R.: Two-generator p-groups of nilpotency class two and their conjugacy classes. Publ. Math. Debrecen, 81, 145–166 (2012)
Beneish, E.: Stable rationality of certain invariant field. J. Algebra, 269, 373–380 (2003)
Bogomolov, F. A.: The Brauer group of quotient spaces by linear group actions. Math. USSR Izv., 30, 455–485 (1988)
Blyth, R., Morse, R.: Computing the nonabelian tensor square of polycyclic groups. J. Algebra, 321, 2139–2148 (2009)
Chen, Y., Ma, R.: Bogomolov multipliers of some groups of order p 6, (preprint available at arXiv: 1302.0584v2 [math.AG]
Garibaldi, S., Merkurjev, A., Serre, J.-P.: “Cohomological invariants in Galois cohomology”, AMS Univ. Lecture Series vol. 28, Amer. Math. Soc., Providence, 2003
Hajja, S., Kang, M.: Some actions of symmetric groups. J. Algebra, 177, 511–535 (1995)
Hoshi, A., Kang, M.: Unramified Brauer groups for groups of order p5, preprint available at arXiv: 1109.2966v1 [math.AC]
Hu, S. J., Kang, M.: Noether’s problem for some p-groups, in “Cohomological and geometric approaches to rationality problems”, edited by F. Bogomolov and Y. Tschinkel, Progress in Math. vol. 282, Birkhäuser, Boston, 2010
Hoshi, A., Kang, M., Kunyavskii, B.: Noether’s problem and unramified Brauer groups. Asian J. Math., 17(4), 689–714 (2013)
Kang, M.: Noether’s problem for metacyclic p-groups. Adv. Math., 203, 554–567 (2005)
Kang, M.: Noether’s problem for p-groups with a cyclic subgroup of index p 2. Adv. Math., 226, 218–234 (2011)
Kang, M.: Bogomolov multipliers and retract rationality for semi-direct products. J. Algebra, 397, 407–425 (2014).
Kang, M., Kunyavskiĭi, B.: The Bogomolov multiplier of rigid finite groups. Archiv Math., 102(3), 209–218 (2014)
Kuniyoshi, H.: On a problem of Chevalley. Nagoya Math. J., 8, 65–67 (1955)
Kunyavskiĭi, B. E.: The Bogomolov multiplier of finite simple groups, Cohomological and geometric approaches to rationality problems, 209217, Progr. Math., 282, Birkhäuser Boston, Inc., Boston, MA, 2010
Michailov, I.: Noether’s problem for abelian extensions of cyclic p-groups. Pacific J. Math, 270(1), 167–189 (2014)
Moravec, P.: Unramified Brauer groups of finite and infinite groups. Amer. J. Math., 134, 1679–1704 (2012)
Moravec, P.: Groups of order p 5 and their unramified Brauer groups. J. Algebra, 372, 420–427 (2012)
Moravec, P.: Unramified Brauer groups and isoclinism. Ars Math. Contemp., 7(2), 337–340 (2014)
Saltman, D. J.: Generic Galois extensions and problems in field theory. Adv. Math., 43, 250–283 (1982)
Saltman, D. J.: Noether’s problem over an algebraically closed field. Invent. Math., 77, 71–84 (1984)
Šafarevic, I. R.: The Lüroth problem. Proc. Steklov Inst. Math., 183, 241–246 (1991)
Swan, R.: Noether’s problem in Galois theory, in “Emmy Noether in Bryn Mawr”, edited by B. Srinivasan and J. Sally, Springer-Verlag, Berlin, 1983
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Michailov, I. Bogomolov multipliers for some p-groups of nilpotency class 2. Acta. Math. Sin.-English Ser. 32, 541–552 (2016). https://doi.org/10.1007/s10114-016-3667-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-016-3667-8
Keywords
- Bogomolov multiplier
- Noether’s problem
- rationality problem
- central product of groups
- p-groups of nilpotency class 2