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Given a finite group   G   and a field   F  , a   G  ‐set   X   gives rise to an    F  [  G  ]   ‐permutation module    F  [  X  ]   . This defines a map from the Burnside ring of   G   to its representation ring over   F  . It is an old problem in representation theory, with wide‐ranging applications in algebra, number theory, and geometry, to give explicit generators of the kernelK  F    (  G  )of this map, that is, to classify pairs of   G  ‐sets   X  ,   Y   such that    F  [  X  ]  ≅  F  [  Y  ]   . When   F   has characteristic 0, a complete description ofK  F    (  G  )is now known. In this paper, we give a similar description ofK  F    (  G  )when   F   is a field of characteristic    p  >  0    in all but the most complicated case, which is when   G   has a subquotient that is a non‐  p  ‐hypo‐elementary    (  p  ,  p  )   ‐Dress group.

Relations between permutation representations in positive characteristic