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Given a      Z [ G ] Z[G]       module  A , we will say a simply connected CW complex  X  is of type ( A, n ) if  X  admits a cellular  G  action, and      H ~  i ( X ) = 0 , i ≠  n , H n ( X ) ≃  A {\tilde H_i}(X) = 0,i \ne n,{H_n}(X) \simeq A       as      Z [ G ] Z[G]       modules. In [5], R. Swan considers the problem posed by Steenrod of whether or not there are finite complexes of type ( A, n ) for all finitely generated  A  and finite  G . Using an invariant defined in terms of      G 0 ( Z [ G ] ) {G_0}(Z[G])      , solutions were obtained for      A = Z p A = {Z_p}       ( p -prime) and      G ⊆  Aut ( Z p ) G \subseteq \operatorname {Aut}\;({Z_p})      . The question of infinite complexes of type ( A, n ) was left open. In this paper we obtain the following complete solution for      Z [ Z p ] Z[{Z_p}]       modules: There are complexes of type      ( A , n ) ( n ⩾  3 ) (A,n)\;(n \geqslant 3)      , and there are finite complexes of type ( A, n ) if and only if the invariant which corresponds to Swan’s invariant for these modules vanishes.

A solution of a problem of Steenrod for cyclic groups of prime order