Group cohomology and control of p-fusion

• 成果类型: Article
• 署名作者: Benson, David J.; Grodal, Jesper; Henke, Ellen
• 署名单位: University of Aberdeen; University of Copenhagen
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-013-0489-5
• 发表日期: 2014
• 页码: 491-507
• 关键词: isomorphisms; characters; spectrum; MAPS

摘要

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.