Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
commutativity-preserving exterior product; ${\widetilde {B}_0}$-pairing; curly exterior square; Bogomolov multiplier
commutativity-preserving exterior product; ${\widetilde {B}_0}$-pairing; curly exterior square; Bogomolov multiplier
Summary:
We conduct an in-depth investigation into the structure of the Bogomolov multiplier for groups of order $p^7$ $(p > 2)$ and exponent $p$. We present a comprehensive classification of these groups, identifying those with nontrivial Bogomolov multipliers and distinguishing them from groups with trivial multipliers. Our analysis not only clarifies the conditions under which the Bogomolov multiplier is nontrivial but also refines existing computational methods, enhancing the process of determining these multipliers for the specified class of $p$-groups.
References:
[1] Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc., III. Ser. 25 (1972), 75-95. DOI 10.1112/PLMS/S3-25.1.75 | MR 0321934 | Zbl 0244.14017
[2] Blyth, R. D., Morse, R. F.: Computing the nonabelian tensor squares of polycyclic groups. J. Algebra 321 (2009), 2139-2148. DOI 10.1016/j.jalgebra.2008.12.029 | MR 2501513 | Zbl 1195.20035
[3] Bogomolov, F. A.: The Brauer group of quotient spaces by linear group actions. Math. USSR, Izv. 30 (1988), 455-485. DOI 10.1070/IM1988v030n03ABEH001024 | MR 0903621 | Zbl 0679.14025
[4] Chen, Y., Ma, R.: Bogomolov multipliers of some groups of order $p^6$. Commun. Algebra 49 (2021), 242-255. DOI 10.1080/00927872.2020.1797074 | MR 4193627 | Zbl 1459.13007
[5] Chu, H., Hu, S.-J., Kang, M.-C., Kunyavskii, B. E.: Noether's problem and the unramified Brauer groups for groups of order 64. Int. Math. Res. Not. 12 (2010), 2329-2366. DOI 10.1093/imrn/rnp217 | MR 2652224 | Zbl 1196.12005
[6] Chu, H., Kang, M.-C.: Rationality of $p$-group actions. J. Algebra 237 (2001), 673-690. DOI 10.1006/jabr.2000.8615 | MR 1816710 | Zbl 1023.13007
[7] Eick, B., Nickel, W.: Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group. J. Algebra 320 (2008), 927-944. DOI 10.1016/j.jalgebra.2008.02.041 | MR 2422322 | Zbl 1163.20022
[8] Hoshi, A., Kang, M.-C.: Unramified Brauer groups for groups of order $p^5$. Available at https://arxiv.org/abs/1109.2966 (2011), 14 pages. DOI 10.48550/arXiv.1109.2966
[9] Jezernik, U., Moravec, P.: Bogomolov multipliers of groups of order 128. Exp. Math. 23 (2014), 174-180. DOI 10.1080/10586458.2014.886980 | MR 3223772 | Zbl 1307.13010
[10] Kunyavskii, B.: The Bogomolov multiplier of finite simple groups. Cohomological and Geometric Approaches to Rationality Problems Progres in Mathematics 282. Birkhäuser, Boston (2010), 209-217. DOI 10.1007/978-0-8176-4934-0_8 | MR 2605170 | Zbl 1204.14006
[11] Michailov, I.: Bogomolov multipliers for some $p$-groups of nilpotency class 2. Acta Math. Sin., Engl. Ser. 32 (2016), 541-552. DOI 10.1007/s10114-016-3667-8 | MR 3483925 | Zbl 1346.14037
[12] Miller, C.: The second homology group of a group; relations among commutators. Proc. Am. Math. Soc. 3 (1952), 588-595. DOI 10.1090/S0002-9939-1952-0049191-5 | MR 0049191 | Zbl 0047.25703
[13] Moravec, P.: Groups of order $p^5$ and their unramified Brauer groups. J. Algebra 372 (2012), 420-427. DOI 10.1016/j.jalgebra.2012.10.002 | MR 2990018 | Zbl 1303.13010
[14] Moravec, P.: Unramified Brauer groups of finite and infinite groups. Am. J. Math. 134 (2012), 1679-1704. DOI 10.1353/ajm.2012.0046 | MR 2999292 | Zbl 1346.20072
[15] Moravec, P.: Unramified Brauer groups and isoclinism. ARS Math. Contemp. 7 (2014), 337-340. DOI 10.26493/1855-3974.392.9fd | MR 3240441 | Zbl 1327.14099
[16] Noether, E.: Gleichungen mit vorgeschriebener Gruppe. Math. Ann. 78 (1917), 221-229 German \99999JFM99999 46.0135.01. DOI 10.1007/BF01457099 | MR 1511893
[17] O'Brien, E.: Polycyclic group. Available at www.math.auckland.ac.nz/ {obrien/GAC-lectures.pdf} (2010), 51 pages.
[18] Saltman, D. J.: Noether's problem over an algebraically closed field. Invent. Math. 77 (1984), 71-84. DOI 10.1007/BF01389135 | MR 0751131 | Zbl 0546.14014
[19] Shafarevich, I. R.: The Lüroth's problem. Proc. Steklov Inst. Math. 183 (1991), 241-246. MR 1092032 | Zbl 0731.14035
[20] Swan, R. G.: Noether's problem in Galois theory. Emmy Noether in Bryn Mawr Springer, New York (1983), 21-40. DOI 10.1007/978-1-4612-5547-5_2 | MR 0713790 | Zbl 0538.12012
[21] Wilkinson, D.: The groups of exponent $p$ and order $p^7$ ($p$ any prime). J. Algebra 118 (1988), 109-119. DOI 10.1016/0021-8693(88)90051-8 | MR 0961329 | Zbl 0651.20025
Search
Partner of
