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A CONSTRUCTION OF TWO DISTINCT CANONICAL SETS OF LIFTS OF BRAUER CHARACTERS OF A p-SOLVABLE GROUP (2006)
| Citations: | 4 - 4 self |
BibTeX
@MISC{Cossey06aconstruction,
author = {James P. Cossey},
title = {A CONSTRUCTION OF TWO DISTINCT CANONICAL SETS OF LIFTS OF BRAUER CHARACTERS OF A p-SOLVABLE GROUP},
year = {2006}
}
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Abstract
Abstract. In [5], Navarro defines 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 IBrp(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, 1Q). In this note we give a counterexample to show that this is not the case when 2 ̸ ∈ π. It is known that if N ⊳ G and χ ∈ Bπ(G), then the constituents of χN are in Bπ(N). However, we use the same counterexample to show that if N ⊳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 ∈ Irr(N) inherits this property. 1. Introduction. Let p be a prime, G a finite p-solvable group, and for a class function α of G, let α0 denote the restriction of α to the p-regular elements of G. The celebrated Fong-Swan Theorem asserts that if ϕ is an irreducible Brauer character of G for the prime p, then there necessarily exists an ordinary irreducible character χ of G such
