### Abstract

Let p be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p, and, if p∈. {3, 5}, the degree of every irreducible character in the principal p-block of G is coprime to p. This gives a complete solution to a problem posed by R. Brauer in 1963.

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

Pages (from-to) | 3-11 |

Number of pages | 9 |

Journal | Journal of Algebra |

Volume | 421 |

DOIs | |

State | Published - Jan 1 2015 |

### Fingerprint

### Keywords

- Abelian Sylow subgroups

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*421*, 3-11. https://doi.org/10.1016/j.jalgebra.2014.08.012

**Abelian Sylow subgroups in a finite group, II.** / Navarro, Gabriel; Solomon, Ronald; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 421, pp. 3-11. https://doi.org/10.1016/j.jalgebra.2014.08.012

}

TY - JOUR

T1 - Abelian Sylow subgroups in a finite group, II

AU - Navarro, Gabriel

AU - Solomon, Ronald

AU - Tiep, Pham Huu

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Let p be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p, and, if p∈. {3, 5}, the degree of every irreducible character in the principal p-block of G is coprime to p. This gives a complete solution to a problem posed by R. Brauer in 1963.

AB - Let p be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p, and, if p∈. {3, 5}, the degree of every irreducible character in the principal p-block of G is coprime to p. This gives a complete solution to a problem posed by R. Brauer in 1963.

KW - Abelian Sylow subgroups

UR - http://www.scopus.com/inward/record.url?scp=84908577355&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908577355&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2014.08.012

DO - 10.1016/j.jalgebra.2014.08.012

M3 - Article

VL - 421

SP - 3

EP - 11

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -