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Navarro vertices and normal subgroups in groups of odd order
"... Abstract. Let p be a prime and suppose G is a finite solvable group and χ is an ordinary irreducible character of G. Navarro has shown that one can associate to χ a pair (Q, δ), where Q is a psubgroup of G and δ is an irreducible character of Q, which is unique up to conjugacy. This pair is called ..."
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Abstract. Let p be a prime and suppose G is a finite solvable group and χ is an ordinary irreducible character of G. Navarro has shown that one can associate to χ a pair (Q, δ), where Q is a psubgroup of G and δ is an irreducible character of Q, which is unique up to conjugacy. This pair is called the Navarro vertex of χ, and in this paper we study the behavior of the Navarro vertices of χ with respect to normal subgroups of G when G is assumed to have odd order and χ is a lift of an irreducible Brauer character of G. For NCG, we use these results to give a sufficient condition for the constituents of χN to be lifts of Brauer characters, and we develop a Cliffordtype correspondence for lifts with a given Navarro vertex. 1.
Induction and restriction of lifts of Brauer characters in groups of odd order
"... The FongSwan theorem shows that if ϕ is any irreducible Brauer character of a finite solvable group G, then there necessarily exists an ordinary irreducible character χ of G that restricts to ϕ. In this case we say that χ is a lift of ϕ. If N G and χ ∈ Irr(G) is a lift, it is also known that the c ..."
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The FongSwan theorem shows that if ϕ is any irreducible Brauer character of a finite solvable group G, then there necessarily exists an ordinary irreducible character χ of G that restricts to ϕ. In this case we say that χ is a lift of ϕ. If N G and χ ∈ Irr(G) is a lift, it is also known that the constituents of χN need not be lifts. In this paper we give a sufficient condition for the constituents of χN to be lifts if G is assumed to have odd order. This condition depends on the local structure of G, and in particular the Navarro vertices. We also give sufficient conditions for the constituents of ψG to be lifts if N G and ψ ∈ Irr(N) is a lift. Finally, we will use these results to prove the existence and uniqueness of lifts that behave well with respect to normal subgroups.