  A conjecture of Moore claims that if G is a group and H a finite index
subgroup of G such that G - H has no elements of prime order (e.g. G is torsion
free), then a G-module which is projective over H is projective over G. The
conjecture is known for finite groups. In that case, it is a direct consequence
of Chouinard's theorem which is based on a fundamental result of Serre on the
vanishing of products of Bockstein operators. It was observed by Benson, using
a construction of Baumslag, Dyer and Heller, that the analog of Serre's Theorem
for infinite groups is not true in general. We prove that the conjecture is
true for groups which satisfy the analog of Serre's theorem. Using a result of
Benson and Goodearl, we prove that the conjecture holds for all groups inside
Kropholler's hierarchy LHF, extending a result of Aljadeff, Cornick, Ginosar,
and Kropholler. We show two closure properties for the class of pairs of groups
(G,H) which satisfy the conjecture, the one is closure under morphisms, and the
other is a closure operation which comes from Kropholler's construction. We use
this in order to exhibit cases in which the analog of Serre's theorem does not
hold, and yet the conjecture is true. We will show that in fact there are pairs
of groups (G,H) in which H is a perfect normal subgroup of prime index in G,
and the conjecture is true for (G,H). Moreover, we will show that it is enough
to prove the conjecture for groups of this kind only.
