By Mark L. Lewis, Gabriel Navarro, Donald S. Passman, Thomas R. Wolf
Read Online or Download Character Theory of Finite Groups: Conference in Honor of I. Martin Isaacs, June 3-5, 2009, Universitat De Valencia, Valencia, Spain PDF
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26] Let G be a group of odd order, and let Q be a p-subgroup of G and δ ∈ Irr(Q). Then there exists a well-deﬁned injection χ → χ∗ from the set Irr(G|Q, δ) into rdz(Gδ |δ). To apply Navarro’s star map to the study of lifts, we need to understand the behavior of lifts with respect to the star map . There are two key observations here. First, suppose that χ ∈ Irr(G|Q, δ) is a lift of ϕ ∈ IBrp (G). Then χ∗ ∈ rdz(Gδ |δ) is a lift, and in fact (χ∗ )NG (Q) is a lift of a uniquely deﬁned character ϕ ∈ IBrp (NG (Q)|Q).
Note that the above theorem does not say that the constituents of χN have to be lifts of Brauer characters. In fact, the constituents of χN need not be lifts of Brauer characters. The above theorem is also not true if one removes either the assumption that G has odd order or the assumption that χ is a lift. 4.  Let G be a group of odd order, and suppose that χ ∈ Irr(G) is a lift of a Brauer character. If N G is such that G/N is a p-group, then the constituents of χN are lifts. Moreover, if ψ ∈ Irr(N ) is a constituent of χN such that ψ o = θ, then Gψ = Gθ .
Thus a given character χ ∈ Irr(G) could have many nuclei, and the nuclei might not all be conjugate in G. We now deﬁne the vertex pair that corresponds to a nucleus. 5. Suppose G is π-separable and χ ∈ Irr(G). Let Q be a π subgroup of G and suppose δ ∈ Irr(Q). We say that the pair (Q, δ) is a vertex pair for χ if there exists a nucleus (U, αβ) for χ such that Q is a Hall π -subgroup of U and βQ = δ, where β is the π -special factor of αβ. If (Q, δ) is a vertex pair for χ and δ is linear, we say that (Q, δ) is a linear vertex pair for χ.