2006/05/31 by James P. Cossey, Cossey, James P.
Mathematics · #20C15 #20C20 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20C15 #msc:20C20
paper · pdf · doi:10.48550/arxiv.math/0605782
arxiv created 2006/05/31 · arxiv updated 2009/12/01
Navarro defined the set Irr(G | Q, δ) ⊆ 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 \textupIBrp(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 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 \bpig = Irr(G | Q, 1Q). In this note we give a counterexample to show that this is not the case when 2 \not∈ π. It is known that if N \nrml G and χ∈ \bpig, then the constituents of χN are in \bpi(N). However, we use the same counterexample to show that if N \nrml G, and χ∈ Irr(G| Q, 1Q) is such that θ∈ Irr(N) and [θ, χN] ≠ 0, then it is not necessarily the case that θ∈ \textupIrr(N) inherits this property.