Startseite Invariance of the Schur multiplier, the Bogomolov multiplier and the minimal number of generators under a variant of isoclinism
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Invariance of the Schur multiplier, the Bogomolov multiplier and the minimal number of generators under a variant of isoclinism

  • Ammu E. Antony , Sathasivam Kalithasan und Viji Z. Thomas EMAIL logo
Veröffentlicht/Copyright: 28. November 2023

Abstract

We introduce the 𝑞-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under 𝑞-isoclinism. We prove that the 𝑞-Schur multiplier is invariant under 𝑞-exterior isoclinism, and as an easy consequence, we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if 𝐺 and 𝐻 are 𝑝-groups with G / Z ( G ) H / Z ( H ) , then the cardinalities of the minimal number of generators of 𝐺 and 𝐻 are the same. Moreover, we prove some structural results about non-abelian 𝑞-tensor square of groups.

Award Identifier / Grant number: MTR/2020/000483

Funding statement: Viji Z. Thomas acknowledges research support from SERB, DST, Government of India grant MTR/2020/000483. Sathasivam K. acknowledges the Ministry of Education, Government of India, for the doctoral fellowship under the Prime Minister’s Research Fellows (PMRF) scheme.

Acknowledgements

We thank the anonymous referees for their valuable comments which improved the exposition greatly.

  1. Communicated by: Bettina Eick

References

[1] A. E. Antony, G. Donadze, V. P. Sivaprasad and V. Z. Thomas, The second stable homotopy group of the Eilenberg–Maclane space, Math. Z. 287 (2017), no. 3–4, 1327–1342. 10.1007/s00209-017-1870-7Suche in Google Scholar

[2] R. Baer, Groups with preassigned central and central quotient group, Trans. Amer. Math. Soc. 44 (1938), no. 3, 387–412. 10.1090/S0002-9947-1938-1501973-3Suche in Google Scholar

[3] F. R. Beyl, U. Felgner and P. Schmid, On groups occurring as center factor groups, J. Algebra 61 (1979), no. 1, 161–177. 10.1016/0021-8693(79)90311-9Suche in Google Scholar

[4] R. D. Blyth, F. Fumagalli and M. Morigi, Some structural results on the non-abelian tensor square of groups, J. Group Theory 13 (2010), no. 1, 83–94. 10.1515/jgt.2009.032Suche in Google Scholar

[5] F. Bogomolov and C. Böhning, Isoclinism and stable cohomology of wreath products, Birational Geometry, Rational Curves, and Arithmetic, Simons Symp., Springer, Cham (2013), 57–76. 10.1007/978-1-4614-6482-2_3Suche in Google Scholar

[6] R. Brown, 𝑄-perfect groups and universal 𝑄-central extensions, Publ. Mat. 34 (1990), no. 2, 291–297. 10.5565/PUBLMAT_34290_08Suche in Google Scholar

[7] R. Brown, D. L. Johnson and E. F. Robertson, Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987), no. 1, 177–202. 10.1016/0021-8693(87)90248-1Suche in Google Scholar

[8] R. Brown and J.-L. Loday, Excision homotopique en basse dimension, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 15, 353–356. Suche in Google Scholar

[9] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311–335. 10.1016/0040-9383(87)90004-8Suche in Google Scholar

[10] T. P. Bueno and N. R. Rocco, On the 𝑞-tensor square of a group, J. Group Theory 14 (2011), no. 5, 785–805. 10.1515/jgt.2010.084Suche in Google Scholar

[11] D. Conduché and C. Rodríguez-Fernández, Nonabelian tensor and exterior products modulo 𝑞 and universal 𝑞-central relative extension, J. Pure Appl. Algebra 78 (1992), no. 2, 139–160. 10.1016/0022-4049(92)90092-TSuche in Google Scholar

[12] G. J. Ellis, Tensor products and 𝑞-crossed modules, J. Lond. Math. Soc. (2) 51 (1995), no. 2, 243–258. 10.1112/jlms/51.2.243Suche in Google Scholar

[13] G. J. Ellis and C. Rodríguez-Fernández, An exterior product for the homology of groups with integral coefficients modulo 𝑝, Cah. Topol. Géom. Différ. Catég. 30 (1989), no. 4, 339–343. Suche in Google Scholar

[14] P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141. 10.1515/crll.1940.182.130Suche in Google Scholar

[15] B. Kunyavskiĭ, The Bogomolov multiplier of finite simple groups, Cohomological and Geometric Approaches to Rationality Problems, Progr. Math. 282, Birkhäuser, Boston (2010), 209–217. 10.1007/978-0-8176-4934-0_8Suche in Google Scholar

[16] P. Moravec, Unramified Brauer groups and isoclinism, Ars Math. Contemp. 7 (2014), no. 2, 337–340. 10.26493/1855-3974.392.9fdSuche in Google Scholar

[17] N. R. Rocco and E. C. P. Rodrigues, The 𝑞-tensor square of finitely generated nilpotent groups, 𝑞 odd, J. Algebra Appl. 16 (2017), no. 11, Article ID 1750211. 10.1142/S0219498817502115Suche in Google Scholar

Received: 2023-05-03
Revised: 2023-09-19
Published Online: 2023-11-28
Published in Print: 2024-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 27.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jgth-2023-0066/pdf?lang=de
Button zum nach oben scrollen