Bounds on the number of lifts of a Brauer character in a p-solvable group

The Fong–Swan theorem shows that for a p-solvable group G and Brauer character φ∈IBrp(G), there is an ordinary character χ∈Irr(G) such that χ0=φ, where 0 denotes restriction to the p-regular elements of G. This still holds in the generality of π-separable groups, where IBrp(G) is replaced by Iπ(G). For φ∈Iπ(G), let Lφ={χ∈Irr(G)|χ0=φ}. In this paper we give a lower bound for the size of Lφ in terms of the structure of the normal nucleus of φ and, if G is assumed to be odd and π={p′}, we give an upper bound for Lφ in terms of the vertex subgroup for φ.