### Abstract

Let Z be a normal subgroup of a nite group G, let λ ε Irr(Z) be an irreducible complex character of Z, and let p be a prime number. If p does not divide the integers x(1)/λ(1) for all x ε Irr(G) lying over λ then we prove that the Sylow p-subgroups of G=Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary nite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture.

Original language | English (US) |
---|---|

Pages (from-to) | 1135-1171 |

Number of pages | 37 |

Journal | Annals of Mathematics |

Volume | 178 |

Issue number | 3 |

DOIs | |

State | Published - 2013 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Mathematics*,

*178*(3), 1135-1171. https://doi.org/10.4007/annals.2013.178.3.7

**Characters of relative p'-degree over normal subgroups.** / Navarro, Gabriel; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Annals of Mathematics*, vol. 178, no. 3, pp. 1135-1171. https://doi.org/10.4007/annals.2013.178.3.7

}

TY - JOUR

T1 - Characters of relative p'-degree over normal subgroups

AU - Navarro, Gabriel

AU - Tiep, Pham Huu

PY - 2013

Y1 - 2013

N2 - Let Z be a normal subgroup of a nite group G, let λ ε Irr(Z) be an irreducible complex character of Z, and let p be a prime number. If p does not divide the integers x(1)/λ(1) for all x ε Irr(G) lying over λ then we prove that the Sylow p-subgroups of G=Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary nite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture.

AB - Let Z be a normal subgroup of a nite group G, let λ ε Irr(Z) be an irreducible complex character of Z, and let p be a prime number. If p does not divide the integers x(1)/λ(1) for all x ε Irr(G) lying over λ then we prove that the Sylow p-subgroups of G=Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary nite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture.

UR - http://www.scopus.com/inward/record.url?scp=84884295519&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884295519&partnerID=8YFLogxK

U2 - 10.4007/annals.2013.178.3.7

DO - 10.4007/annals.2013.178.3.7

M3 - Article

AN - SCOPUS:84884295519

VL - 178

SP - 1135

EP - 1171

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 3

ER -