Arkiv för Matematik

  • Ark. Mat.
  • Volume 52, Number 2 (2014), 203-225.

Nilpotent $p\mspace{1mu}$-local finite groups

José Cantarero, Jérôme Scherer, and Antonio Viruel

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We provide characterizations of $p\mspace {1mu}$-nilpotency for fusion systems and $p\mspace {1mu}$-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.


The first author was partially supported by FEDER/MEC grant MTM2010-20692. The second author was partially supported by FEDER/MEC grant MTM2010-20692 and the Max Planck Institute, Bonn. The third author was partially supported by FEDER/MCI grant MTM2010-18089, and Junta de Andalucía grants FQM-0213 and P07-FQM-2863.

Article information

Ark. Mat., Volume 52, Number 2 (2014), 203-225.

Received: 18 May 2012
Revised: 22 October 2012
First available in Project Euclid: 30 January 2017

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2013 © Institut Mittag-Leffler


Cantarero, José; Scherer, Jérôme; Viruel, Antonio. Nilpotent $p\mspace{1mu}$-local finite groups. Ark. Mat. 52 (2014), no. 2, 203--225. doi:10.1007/s11512-013-0181-4.

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  • Alperin, J. L., Centralizers of abelian normal subgroups of $p\mspace {1mu}$-groups, J. Algebra 1 (1964), 110–113.
  • Benson, D. J. and Smith, S. D., Classifying Spaces of Sporadic Groups, Mathematical Surveys and Monographs 147, Amer. Math. Soc., Providence, RI, 2008.
  • Blackburn, N., Automorphisms of finite $p\mspace {1mu}$-groups, J. Algebra 3 (1966), 28–29.
  • Bousfield, A. K. and Kan, D. M., Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer, Berlin–Heidelberg, 1972.
  • Broto, C., Castellana, N., Grodal, J., Levi, R. and Oliver, B., Subgroup families controlling $p\mspace {1mu}$-local finite groups, Proc. Lond. Math. Soc. 91 (2005), 325–354.
  • Broto, C., Castellana, N., Grodal, J., Levi, R. and Oliver, B., Extensions of $p\mspace {1mu}$-local finite groups, Trans. Amer. Math. Soc. 359 (2007), 3791–3858.
  • Broto, C., Levi, R. and Oliver, B., Homotopy equivalences of $p\mspace {1mu}$-completed classifying spaces of finite groups, Invent. Math. 151 (2003), 611–664.
  • Broto, C., Levi, R. and Oliver, B., The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856.
  • Brunetti, M., A new cohomological criterion for the $p\mspace {1mu}$-nilpotence of groups, Canad. Math. Bull. 41 (1998), 20–22.
  • Castellana, N. and Morales, D., Vector bundles over classifying spaces of $p\mspace {1mu}$-local finite groups, Preprint, 2010.
  • Chermak, A., Fusion systems and localities, Preprint, 2011.
  • Craven, D. A., The Theory of Fusion Systems, Cambridge Studies in Advanced Mathematics 131, Cambridge University Press, Cambridge, 2011.
  • Díaz, A., Glesser, A., Mazza, N. and Park, S., Glauberman’s and Thompson’s theorems for fusion systems, Proc. Amer. Math. Soc. 137 (2009), 495–503.
  • Díaz, A., Glesser, A., Park, S. and Stancu, R., Tate’s and Yoshida’s theorem on control of transfer for fusion systems, J. Lond. Math. Soc. 84 (2011), 475–494.
  • Farjoun, E. D., Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Math. 1622, Springer, Berlin–Heidelberg, 1996.
  • Glauberman, G., Global and local properties of finite groups, in Finite Simple Groups (Oxford, 1969), pp. 1–64, Academic Press, London, 1971.
  • Glesser, A., Sparse fusion systems, Proc. Edinb. Math. Soc. 56 (2013), 135–150.
  • González-Sánchez, J., A $p\mspace {1mu}$-nilpotency criterion, Arch. Math. (Basel) 94 (2010), 201–205.
  • Gorenstein, D., Finite Groups, 2nd ed., Chelsea, New York, 1980.
  • Hopkins, M. J., Kuhn, N. J. and Ravenel, D. C., Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553–594.
  • Huppert, B., Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften 134, Springer, Berlin, 1967.
  • Isaacs, I. M., Algebra, Brooks/Cole, Pacific Grove, 1994.
  • Kessar, R. and Linckelmann, M., A block theoretic analogue of a theorem of Glauberman and Thompson, Proc. Amer. Math. Soc. 131 (2003), 35–40.
  • Kessar, R. and Linckelmann, M., ZJ-theorems for fusion systems, Trans. Amer. Math. Soc. 360 (2008), 3093–3106.
  • Kochman, S. O., Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs 7, Amer. Math. Soc., Providence, RI, 1996.
  • Levi, R. and Oliver, B., Construction of 2-local finite groups of a type studied by Solomon and Benson, Geom. Topol. 6 (2002), 917–990.
  • Linckelmann, M., Introduction to fusion systems, in Group Representation Theory, pp. 79–113, EPFL Press, Lausanne, 2007.
  • Mislin, G., Localization with respect to K-theory, J. Pure Appl. Algebra 10 (1977/78), 201–213.
  • Oliver, B., Equivalences of classifying spaces completed at odd primes, Math. Proc. Cambridge Philos. Soc. 137 (2004), 321–347.
  • Oliver, B., Equivalences of Classifying Spaces Completed at the Prime Two, Mem. Amer. Math. Soc. 180:848, 2006.
  • Oliver, B., Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory, Preprint, 2012.
  • Puig, L., Full Frobenius systems and their localizing categories, Preprint, 2001.
  • Quillen, D., A cohomological criterion for $p\mspace {1mu}$-nilpotence, J. Pure Appl. Algebra 1 (1971), 361–372.
  • Robinson, D. J. S., A Course in the Theory of Groups, Graduate Texts in Mathematics 80, Springer, New York, 1982.
  • Stammbach, U., Another homological characterisation of finite $p\mspace {1mu}$-nilpotent groups, Math. Z. 156 (1977), 209–210.
  • Tate, J., Nilpotent quotient groups, Topology 3 (1964), 109–111.
  • Wilson, W. S., K(n+1) equivalence implies K(n) equivalence, in Homotopy Invariant Algebraic Structures (Baltimore, MD, 1998), Contemp. Math. 239, pp. 375–376, Amer. Math. Soc., Providence, RI, 1999.