Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the $ p'$-extensions of inertial blocks


Author: Yuanyang Zhou
Journal: Proc. Amer. Math. Soc. 144 (2016), 41-54
MSC (2010): Primary 20C20
DOI: https://doi.org/10.1090/proc/12691
Published electronically: September 24, 2015
MathSciNet review: 3415575
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p$ be a prime number, $ G$ a finite group, $ H$ a normal subgroup of $ G$, and $ b$ a $ p$-block of $ H$. Assuming that the index of $ H$ in $ G$ is coprime to $ p$, we prove that any $ p$-block of $ G$ covering $ b$ is inertial if and only if the block $ b$ is inertial.


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

  • [1] Michel Broué and Lluis Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), no. 2, 117-128. MR 558864 (81d:20011), https://doi.org/10.1007/BF01392547
  • [2] Everett C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), no. 3, 241-262. MR 593823 (82c:16028), https://doi.org/10.1007/BF01161413
  • [3] Shigeo Koshitani and Naoko Kunugi, Broué's conjecture holds for principal 3-blocks with elementary abelian defect group of order 9, J. Algebra 248 (2002), no. 2, 575-604. MR 1882112 (2002j:20021), https://doi.org/10.1006/jabr.2001.9048
  • [4] Burkhard Külshammer and Lluis Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), no. 1, 17-71. MR 1069239 (91i:20009), https://doi.org/10.1007/BF01233419
  • [5] Andrei Marcus, On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra 184 (1996), no. 2, 372-396. MR 1409219 (97m:20012), https://doi.org/10.1006/jabr.1996.0265
  • [6] Lluis Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), no. 2, 265-292. MR 607966 (82d:20015), https://doi.org/10.1007/BF01261873
  • [7] Lluis Puig, Local fusions in block source algebras, J. Algebra 104 (1986), no. 2, 358-369. MR 866781 (88c:20020), https://doi.org/10.1016/0021-8693(86)90221-8
  • [8] Lluis Puig, Pointed groups and construction of modules, J. Algebra 116 (1988), no. 1, 7-129. MR 944149 (89e:20024), https://doi.org/10.1016/0021-8693(88)90195-0
  • [9] Lluis Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), no. 1, 77-116. MR 943924 (89e:20023), https://doi.org/10.1007/BF01393688
  • [10] Lluis Puig, On the local structure of Morita and Rickard equivalences between Brauer blocks, Progress in Mathematics, vol. 178, Birkhäuser Verlag, Basel, 1999. MR 1707300 (2001d:20006)
  • [11] Lluis Puig, Nilpotent extensions of blocks, Math. Z. 269 (2011), no. 1-2, 115-136. MR 2836062 (2012j:20030), https://doi.org/10.1007/s00209-010-0718-1
  • [12] Lluis Puig and Yuanyang Zhou, Glauberman correspondents and extensions of nilpotent block algebras, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 809-837. MR 2927809, https://doi.org/10.1112/jlms/jdr069
  • [13] Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. (3) 72 (1996), no. 2, 331-358. MR 1367082 (97b:20011), https://doi.org/10.1112/plms/s3-72.2.331
  • [14] Raphaël Rouquier, Block theory via stable and Rickard equivalences, Modular representation theory of finite groups (Charlottesville, VA, 1998), de Gruyter, Berlin, 2001, pp. 101-146. MR 1889341 (2003g:20018)
  • [15] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237 (82e:12016)
  • [16] Jacques Thévenaz, $ G$-algebras and modular representation theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1365077 (96j:20017)
  • [17] Yuanyang Zhou, A remark on Rickard complexes, J. Algebra 399 (2014), 845-853. MR 3144614, https://doi.org/10.1016/j.jalgebra.2013.10.015

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20C20

Retrieve articles in all journals with MSC (2010): 20C20


Additional Information

Yuanyang Zhou
Affiliation: Department of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China
Email: zhouyuanyang@mail.ccnu.edu.cn

DOI: https://doi.org/10.1090/proc/12691
Received by editor(s): January 9, 2014
Received by editor(s) in revised form: December 5, 2014
Published electronically: September 24, 2015
Additional Notes: The author was supported by self-determined research funds of CCNU from the colleges’ basic research and operation of MOE (No. 20205140052) and by NSFC (No. 11071091)
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society