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

Abstract.

In [5], Navarro defines the set $\hbox{Irr}(G|Q,\delta)\subseteqq \hbox{Irr}(G)$, where Q is a p-subgroup of a p-solvable group G, and shows that if δ is the trivial character of Q, then Irr(G|Q, δ) provides a set of canonical lifts of IBr _p (G), the irreducible Brauer characters with vertex Q. Previously, in [2], Isaacs defined a canonical set of lifts B_π(G) of I_π(G). Both of these results extend the Fong-Swan Theorem to π-separable groups, and both construct canonical sets of lifts of the generalized Brauer characters. It is known that in the case that 2∈π, or if |G| is odd, we have B_π(G) = Irr(G|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 $N\triangleleft G$ and χ∈B_π(G), then the constituents of χ _N are in B_π (N). However, we use the same counterexample to show that if $N\triangleleft G$ , and χ∈Irr(G|Q, 1 _Q ) is such that θ ∈Irr(N) and [θ, χ _N ]  ≠  0, then it is not necessarily the case that θ ∈Irr(N) inherits this property.