Diagonal resolutions for metacyclic groups

We show the finite metacyclic groups G(p,q) admit a class of projective resolutions which are periodic of period 2q and which in addition possess the properties that a) the differentials are 2×2 diagonal matrices; b) the Swan-Wall finiteness obstruction (cf. [19,20]) vanishes. We obtain thereby a purely algebraic proof of Petrie's Theorem ([14]) that G(p,q) has free period 2q.