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 ${\textup{IBr}}_p(G)$, 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 $2 \not\in \pi$. It is known that if $N \nrml G$ and $\chi \in \bpig$, 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 $[\theta, \chi_N] \neq 0$, then it is not necessarily the case that $\theta \in \textup{Irr}(N)$ inherits this property.