A construction of two distinct canonical sets of lifts of Brauer characters of a p-solvable group

Abstract

Navarro defined the set $Irr(G\mid Q,\delta )\subseteq Irr(G)$, where $Q$ is a $p$-subgroup of a $p$-solvable group $G$, and shows that if $\delta$ is the trivial character of $Q$, then $Irr(G\mid Q,\delta )$ provides a set of canonical lifts of ${\text{IBr}}_p(G)$, the irreducible Brauer characters with vertex $Q$. Previously, Isaacs defined a canonical set of lifts $\bpig$ of $\ipig$. Both of these results extend the Fong-Swan Theorem to $\pi$-separable groups, and both construct canonical sets of lifts of the generalized Brauer characters. It is known that in the case that $2\in \pi$, or if $|G|$ is odd, we have $\bpig = {Irr}(G \mid Q, 1_Q)$. In this note we give a counterexample to show that this is not the case when $2\notin \pi$. It is known that if $N \nrml G$ and $χ\in \bpig$, then the constituents of ${\chi }_N$ are in $\bpi(N)$. However, we use the same counterexample to show that if $N \nrml G$, and $\chi \in Irr(G\mid Q,1_Q)$ is such that $\theta \in Irr(N)$ and $[\theta ,{\chi }_N]\ne 0$, then it is not necessarily the case that $\theta \in \text{Irr}(N)$ inherits this property.