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Rigidity of $ p$-completed classifying spaces of alternating groups and classical groups over a finite field


Author: Kenshi Ishiguro
Journal: Trans. Amer. Math. Soc. 329 (1992), 697-713
MSC: Primary 55R35; Secondary 55S37, 57T99
DOI: https://doi.org/10.1090/S0002-9947-1992-1096261-9
MathSciNet review: 1096261
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Abstract: A $ p$-adic rigid structure of the classifying spaces of certain finite groups $ \pi $, including alternating groups $ {A_n}$ and finite classical groups, is shown in terms of the maps into the $ p$-completed classifying spaces of compact Lie groups. The spaces $ (B\pi )_p^ \wedge $ have no nontrivial retracts. As an application, it is shown that $ (B{A_n})_p^ \wedge \simeq (B{\Sigma _n})_p^ \wedge $ if and only if $ n\not \equiv 0,1,\;\bmod \,p$. It is also shown that $ (BSL(n,{\mathbb{F}_q}))_p^ \wedge \simeq (BGL(n,{\mathbb{F}_q}))_p^ \wedge $ where $ q$ is a power of $ p$ if and only if $ (n,q - 1) = 1$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1096261-9
Keywords: Classifying space, $ p$-completion, mapping space, simple group, compact Lie group
Article copyright: © Copyright 1992 American Mathematical Society

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