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 Bπ(G) of Iπ(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 Bπ(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∈Bπ(G), then the constituents of χ N are in Bπ (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.