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.