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A      p p      -adic rigid structure of the classifying spaces of certain finite groups      π  \pi      , including alternating groups      A n {A_n}       and finite classical groups, is shown in terms of the maps into the      p p      -completed classifying spaces of compact Lie groups. The spaces      ( B π  ) p ∧  (B\pi )_p^ \wedge       have no nontrivial retracts. As an application, it is shown that      ( B A n ) p ∧  ≃  ( B Σ  n ) p ∧  (B{A_n})_p^ \wedge \simeq (B{\Sigma _n})_p^ \wedge       if and only if      n ≢ 0 , 1 , mod p n\not \equiv 0,1,\;\bmod \,p      . It is also shown that      ( B S L ( n , F q ) ) p ∧  ≃  ( B G L ( n , F q ) ) p ∧  (BSL(n,{\mathbb {F}_q}))_p^ \wedge \simeq (BGL(n,{\mathbb {F}_q}))_p^ \wedge       where      q q       is a power of      p p       if and only if      ( n , q −  1 ) = 1 (n,q - 1) = 1      .

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