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Denote by  ƒ λ  ( G  ) the number of free subgroups of index  λm G   , where  m G   is the least common multiple of the orders of the finite subgroups in  G . The present paper develops a general theory for the  p -divisibility of  ƒ λ  ( G  ), where  p  is a prime dividing  m G  . Among other things, we obtain an explicit combinatorial description of  ƒ λ  ( G  ) modulo  p , leading to an optimal generalisation of Stothers’ explicit formula for the parity of  ƒ λ  (PSL 2  (ℤ)).

Modular arithmetic of free subgroups