Counting lifts of irreducible Brauer characters

Abstract

Let $p$ be an odd prime, and suppose that $G$ is a $p$-solvable group and $\phi \in \mathrm{IBr}(G)$ has vertex $Q$. In 2011, Cossey, Lewis and Navarro proved that the number of lifts of $\phi$ is at most $|Q:Q^{\prime }|$ whenever $Q$ is normal in $G$. In this paper, we present an explicit description of the set of lifts of $\phi$ with a given vertex pair $(Q,\delta )$ under a weaker condition on $Q$, and thus generalize their result.