## Details

Original language | English |
---|---|

Pages (from-to) | 130-141 |

Number of pages | 12 |

Journal | Journal of algebra |

Volume | 587 |

Early online date | 19 Aug 2021 |

Publication status | Published - 1 Dec 2021 |

## Abstract

Let G be a p-solvable group, P≤G a p-subgroup and χ∈Irr(G) such that χ(1)
_{p}≥|G:P|
_{p}. We prove that the restriction χ
_{P} is a sum of characters induced from subgroups Q≤P such that χ(1)
_{p}=|G:Q|
_{p}. This generalizes previous results by Giannelli–Navarro and Giannelli–Sambale on the number of linear constituents of χ
_{P}. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer–Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde.

## Keywords

- Character restriction, Linear constituents, p-solvable groups

## ASJC Scopus subject areas

- Mathematics(all)
**Algebra and Number Theory**

## Cite this

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- BibTeX
- RIS

**Restrictions of characters in p-solvable groups.**/ Rossi, Damiano; Sambale, Benjamin.

In: Journal of algebra, Vol. 587, 01.12.2021, p. 130-141.

Research output: Contribution to journal › Article › Research › peer review

*Journal of algebra*, vol. 587, pp. 130-141. https://doi.org/10.1016/j.jalgebra.2021.07.034

*Journal of algebra*,

*587*, 130-141. https://doi.org/10.1016/j.jalgebra.2021.07.034

}

TY - JOUR

T1 - Restrictions of characters in p-solvable groups

AU - Rossi, Damiano

AU - Sambale, Benjamin

N1 - Funding Information: We thank Gabriel Navarro for sharing his insights on a previous version of this paper, and for requesting more computer checking. We appreciate Eugenio Giannelli's effort to prove corresponding results for symmetric groups. Moreover, Alexander Hulpke has kindly provided an updated database [12] of all perfect groups of order at most 10 6 . Thomas Breuer has introduced the authors to numerous tricks regarding character tables in GAP. The first author is supported by the research training group GRK2240 : Algebro-geometric Methods in Algebra, Arithmetic and Topology of the German Research Foundation . The second author is supported by the German Research Foundation ( SA 2864/1-2 and SA 2864/3-1 ).

PY - 2021/12/1

Y1 - 2021/12/1

N2 - Let G be a p-solvable group, P≤G a p-subgroup and χ∈Irr(G) such that χ(1) p≥|G:P| p. We prove that the restriction χ P is a sum of characters induced from subgroups Q≤P such that χ(1) p=|G:Q| p. This generalizes previous results by Giannelli–Navarro and Giannelli–Sambale on the number of linear constituents of χ P. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer–Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde.

AB - Let G be a p-solvable group, P≤G a p-subgroup and χ∈Irr(G) such that χ(1) p≥|G:P| p. We prove that the restriction χ P is a sum of characters induced from subgroups Q≤P such that χ(1) p=|G:Q| p. This generalizes previous results by Giannelli–Navarro and Giannelli–Sambale on the number of linear constituents of χ P. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer–Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde.

KW - Character restriction

KW - Linear constituents

KW - p-solvable groups

UR - http://www.scopus.com/inward/record.url?scp=85113357274&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2021.07.034

DO - 10.1016/j.jalgebra.2021.07.034

M3 - Article

VL - 587

SP - 130

EP - 141

JO - Journal of algebra

JF - Journal of algebra

SN - 0021-8693

ER -