### Abstract

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito-Michler Theorem asserts that if a prime p does not divide the degree of any χ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irr_{rv}(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr_{rv}(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.

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

Pages (from-to) | 755-774 |

Number of pages | 20 |

Journal | Mathematische Zeitschrift |

Volume | 259 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2008 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*259*(4), 755-774. https://doi.org/10.1007/s00209-007-0247-8

**Primes dividing the degrees of the real characters.** / Dolfi, Silvio; Navarro, Gabriel; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 259, no. 4, pp. 755-774. https://doi.org/10.1007/s00209-007-0247-8

}

TY - JOUR

T1 - Primes dividing the degrees of the real characters

AU - Dolfi, Silvio

AU - Navarro, Gabriel

AU - Tiep, Pham Huu

PY - 2008/8

Y1 - 2008/8

N2 - Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito-Michler Theorem asserts that if a prime p does not divide the degree of any χ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irrrv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irrrv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.

AB - Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito-Michler Theorem asserts that if a prime p does not divide the degree of any χ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irrrv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irrrv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.

UR - http://www.scopus.com/inward/record.url?scp=43749107273&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43749107273&partnerID=8YFLogxK

U2 - 10.1007/s00209-007-0247-8

DO - 10.1007/s00209-007-0247-8

M3 - Article

AN - SCOPUS:43749107273

VL - 259

SP - 755

EP - 774

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 4

ER -