Group cohomology and control of p-fusion

David J. Benson, Jesper Grodal*, Ellen Henke

*Corresponding author for this work

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We show that if an inclusion of finite groups Ha parts per thousand currency signG of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories.

The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.

Original languageEnglish
Pages (from-to)491-507
Number of pages16
JournalInventiones Mathematicae
Volume197
Issue number3
Early online date13 Nov 2013
DOIs
Publication statusPublished - Sep 2014

Keywords

  • classifying-spaces
  • isomorphisms
  • homology
  • ring
  • nilpotency
  • spectrum
  • maps

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