Inductive sources and subnormal subgroups

Abstract

A character pair (H, θ) in a group G is a subgroup H and a character θ ∈ Irr(H). Following Dade, we say that a character pair (H, θ) is an inductive source in G if induction to G defines an injective map from the irreducible characters of the stabilizer of (H, θ) that lie over θ into Irr(G). (By the Clifford correspondence, this necessarily happens if H is normal in G.) A character pair is said to be conjugate stable if its conjugates satisfy a certain technical condition. We show that an inductive source must be conjugate stable and we present an example of a character pair that is conjugate stable but is not an inductive source. Finally, if (H, θ) is conjugate stable and H is a subnormal subgroup of G, we show that (H, θ) must be an inductive source in G.