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

Abstract

The Fong-Swan theorem shows that for a $p$-solvable group $G$ and Brauer character $ϕ\in \ibrg$, there is an ordinary character $χ\in \irrg$ such that ${\chi }^0=\varphi$, where ${}^0$ denotes restriction to the $p$-regular elements of $G$. This still holds in the generality of $\pi$-separable groups \cite{bpi}, where $\ibrg$ is replaced by $\ipig$. For $ϕ\in \ipig$, let $L_ϕ = \{χ\in \irrg \mid χ^0 = ϕ\}$. In this paper we give a lower bound for the size of $L_{\varphi }$ in terms of the structure of the normal nucleus of $\varphi$ and, if $G$ is assumed to be odd and $\pi =\{p^{\prime }\}$, we give an upper bound for $L_{\varphi }$ in terms of the vertex subgroup for $\varphi$.