Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cohomology of uniserial $ p$-adic space groups


Authors: Antonio Díaz Ramos, Oihana Garaialde Ocaña and Jon González-Sánchez
Journal: Trans. Amer. Math. Soc. 369 (2017), 6725-6750
MSC (2010): Primary 20J06, 55T10
DOI: https://doi.org/10.1090/tran/7145
Published electronically: May 5, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A decade ago, J. F. Carlson proved that there are finitely many cohomology rings of finite $ 2$-groups of fixed coclass, and he conjectured that this result ought to be true for odd primes. In this paper, we prove the non-twisted case of Carlson's conjecture for any prime and we show how to proceed in the twisted case.


References [Enhancements On Off] (What's this?)

  • [1] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
  • [2] Jon F. Carlson, Coclass and cohomology, J. Pure Appl. Algebra 200 (2005), no. 3, 251-266. MR 2147269, https://doi.org/10.1016/j.jpaa.2005.01.001
  • [3] Serena Cicalò, Willem A. de Graaf, and Michael Vaughan-Lee, An effective version of the Lazard correspondence, J. Algebra 352 (2012), 430-450. MR 2862197, https://doi.org/10.1016/j.jalgebra.2011.11.031
  • [4] Bettina Eick, Determination of the uniserial space groups with a given coclass, J. London Math. Soc. (2) 71 (2005), no. 3, 622-642. MR 2132374, https://doi.org/10.1112/S0024610705006332
  • [5] Bettina Eick and David J. Green, The Quillen categories of $ p$-groups and coclass theory, Israel J. Math. 206 (2015), no. 1, 183-212. MR 3319637, https://doi.org/10.1007/s11856-014-1138-4
  • [6] Graham Ellis, Cohomological periodicities of crystallographic groups, J. Algebra 445 (2016), 537-544. MR 3418068, https://doi.org/10.1016/j.jalgebra.2015.08.015
  • [7] Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. MR 1144017
  • [8] Gustavo A. Fernández-Alcober, An introduction to finite $ p$-groups: regular $ p$-groups and groups of maximal class, 16th School of Algebra, Part I (Portuguese) (Brasília, 2000), Mat. Contemp. 20 (2001), 155-226. MR 1868828
  • [9] Gustavo A. Fernández-Alcober, Jon González-Sánchez, and Andrei Jaikin-Zapirain, Omega subgroups of pro-$ p$ groups, Israel J. Math. 166 (2008), 393-412. MR 2430441, https://doi.org/10.1007/s11856-008-1036-8
  • [10] Christopher J. Hillar and Darren L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no. 10, 917-923. MR 2363058
  • [11] E. I. Khukhro, $ p$-automorphisms of finite $ p$-groups, London Mathematical Society Lecture Note Series, vol. 246, Cambridge University Press, Cambridge, 1998. MR 1615819
  • [12] C. R. Leedham-Green, The structure of finite $ p$-groups, J. London Math. Soc. (2) 50 (1994), no. 1, 49-67. MR 1277754, https://doi.org/10.1112/jlms/50.1.49
  • [13] C. R. Leedham-Green and W. Plesken, Some remarks on Sylow subgroups of general linear groups, Math. Z. 191 (1986), no. 4, 529-535. MR 832810, https://doi.org/10.1007/BF01162342
  • [14] C. R. Leedham-Green and S. McKay, The structure of groups of prime power order, London Mathematical Society Monographs. New Series, vol. 27, Oxford University Press, Oxford, 2002. MR 1918951
  • [15] C. R. Leedham-Green, S. McKay, and W. Plesken, Space groups and groups of prime power order. VI. A bound to the dimension of a $ 2$-adic space group with fixed coclass, J. London Math. Soc. (2) 34 (1986), no. 3, 417-425. MR 864445, https://doi.org/10.1112/jlms/s2-34.3.417
  • [16] Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Bd. 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
  • [17] J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
  • [18] S. McKay, The precise bound to the coclass of space groups, J. London Math. Soc. (2) 50 (1994), no. 3, 488-500. MR 1299453, https://doi.org/10.1112/jlms/50.3.488
  • [19] Lenny Taelman, Characteristic classes for curves of genus one, Michigan Math. J. 64 (2015), no. 3, 633-654. MR 3394262, https://doi.org/10.1307/mmj/1441116661
  • [20] Thomas S. Weigel, $ p$-central groups and Poincaré duality, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4143-4154. MR 1621710, https://doi.org/10.1090/S0002-9947-99-02385-5
  • [21] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20J06, 55T10

Retrieve articles in all journals with MSC (2010): 20J06, 55T10


Additional Information

Antonio Díaz Ramos
Affiliation: Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Apdo correos 59, 29080 Málaga, Spain
Email: adiazramos@uma.es

Oihana Garaialde Ocaña
Affiliation: Matematika Saila, Euskal Herriko Unibertsitatearen Zientzia eta Teknologia Fakultatea, posta-kutxa 644, 48080 Bilbao, Spain
Email: oihana.garayalde@ehu.es

Jon González-Sánchez
Affiliation: Departamento de Matemáticas, Facultad de Ciencia y Tecnología de la Universidad del Pais Vasco, Apdo correos 644, 48080 Bilbao, Spain
Email: jon.gonzalez@ehu.es

DOI: https://doi.org/10.1090/tran/7145
Received by editor(s): May 10, 2016
Received by editor(s) in revised form: October 12, 2016
Published electronically: May 5, 2017
Additional Notes: The first author was supported by MICINN grant RYC-2010-05663 and partially supported by MEC grant MTM2013-41768-P and Junta de Andalucía grant FQM-213
The second author was supported by the Basque Government Ph.D. grant PRE_2015_2_0130 and partially supported by the Spanish Ministry of Economy and Competitivity grant MTM2014-53810-C2-2-P and by the Basque Government grants IT753-13 and IT974-16
The third author acknowledges the support of grants MTM2011-28229-C02-01 and MTM2014-53810-C2-2-P from the Spanish Ministry of Economy and Competitivity, the Ramon y Cajal Programme of the Spanish Ministry of Science and Innovation, grant RYC-2011-08885, and of the Basque Government, grants IT753-13 and IT974-16
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society