Clifford theory of characters in induced blocks

Abstract

We present a new criterion to predict if a character of a finite group extends. Let G be a finite group and p a prime. For N ⊳G, we consider p-blocks b and b of N and NN (D), respectively, with (b ) = b, where D is a defect group of b. Under the assumption that G coincides with a normal subgroup G[b] of G, which was introduced by Dade early 1970’s, we give a character correspondence between the sets of all irreducible constituents of φ and those of (φ′)NG(D) where φ and φ are irreducible Brauer characters in b and b, respectively. This implies a sort of generalization of the theorem of Harris-Knorr. An important tool is the existence of certain extensions that also helps in checking the inductive Alperin-McKay and inductive Blockwise Alperin Weight conditions, due to the second author.