This paper generalizes a result of Okuyama [4, Theorem 31 on the situation wherein a block of a subgroup of a finite group induces to a block of the group (in the classical sense of Brauer; see [ 1, p. 1361). Our generalization, which comprises Lemma A and Theorem B, is proved mainly by observing that Okuyama’s arguments hold under less stringent hypotheses. Our results are applied in Theorem C to establish a necessary and sufficient condition for when a block of defect zero of a subgroup induces to a block of the group. Another application is Theorem D, which concerns block induction from a normal subgroup. Throughout the paper, H denotes a subgroup of an arbitrary finite group G and p is a fixed prime rational integer. Let R be the ring of integers in a p-adic number field K of characteristic zero, let rt be a generator of J(R), and set R = R/rcR. If M is an RG-module, i@ denotes M/nM. Assume that K and R are splitting fields for all subgroups of G. Let v denote the p-adic valuation on K, scaled so that v(p) = 1. Let v( / G I) = a and r( 1 H 1) = m, so that IGJP=p”, /I1l,,=p”‘. Let B, h denote fixed p-blocks of G (resp. H), where B has defect d Let Irr (B) (resp. Irr(b)), denote the set of ordinary irreducible characters in B (resp. h). If 0 is a rational integral combination of Brauer characters of G, let 8 denote the generalized character of G defined by [ 1, IV.1.21,