Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example,
$\mathbb S$
n ×
$\mathbb S$
2n is a space of type (0, 1) and the one-point union
$\mathbb S$
n ∨
$\mathbb S$
2n ∨
$\mathbb S$
3n is a space of type (0, 0)). It is known that a finite group G that contains ℤp ⊕ ℤp ⊕ ℤp, p a prime, cannot act freely on
$\mathbb S$
n ×
$\mathbb S$
2n. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G cannot contain ℤp ⊕ ℤp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that ℤ2 is the only group that can act freely on X.