Subnormal Inductive Sources and π-Partial Characters

Abstract

A character pair (H,θ) in a group G consists of a subgroup H and a character θ ∈ Irr (H). A character pair (H,θ) is an inductive source in G if induction to G defines an injective map from the irreducible characters of T, the stabilizer of (H,θ), that lie over θ into Irr (G). Let π be a set of primes, and suppose G is π-separable. We consider Isaacs' π-partial characters and their canonical lifts. If θ ∈ Irr (H) is such a lift, then the restriction θ0 of θ to the π-elements of H is an irreducible π-partial character of H. In this paper, when (H,θ) is an inductive source so that H is subnormal and θ is a canonical lift, we show that induction is an injection from the irreducible π-partial characters of T that lie over θ0 to the irreducible π-partial characters of G. We apply this to obtain a partial generalization of Isaacs' nucleus of a character, and present several examples to see what can go wrong with our generalization.