A construction of two distinct canonical sets of lifts of Brauer characters of a p-solvable group |
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Authors: | James P Cossey |
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Institution: | (1) Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA |
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Abstract: | In 5], Navarro defines the set
, 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
. It is known that if
and χ∈Bπ(G), then the constituents of χN are in Bπ (N). However, we use the same counterexample to show that if
, and χ∈Irr(G|Q, 1Q) 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.
Received: 17 October 2005 |
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Keywords: | Primary 20C20 |
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