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Navarro, Gabriel; Tiep, Pham Huu

Characters of relative \(p^\prime\)-degree over normal subgroups. (English) Zbl 1372.20016

Ann. Math. (2) 178, No. 3, 1135-1171 (2013).
Authors’ abstract: Let \(Z\) be a normal subgroup of a finite group \(G\), let \(\lambda \in \mathrm{Irr}(Z)\) be an irreducible complex character of \(Z\), and let \(p\) be a prime number. If \(p\) does not divide the integers \(\chi(1)/\lambda(1)\) for all \(\chi \in \mathrm{Irr}(G)\) lying over \(\lambda\), then we prove that the Sylow \(p\)-subgroups of \(G/Z\) are abelian. This theorem, which generalizes the Gluck-Wolf theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer height zero conjecture.
Reviewer: Morten E. Harris (Chicago)

Cited in 33 Documents

MSC:

20C15 Ordinary representations and characters
20C20 Modular representations and characters
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure

Keywords:

Brauer’s height zero conjecture

Software:

GAP
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\textit{G. Navarro} and \textit{P. H. Tiep}, Ann. Math. (2) 178, No. 3, 1135--1171 (2013; Zbl 1372.20016)
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References:

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