## Details

Originalsprache | Englisch |
---|---|

Aufsatznummer | 3 |

Seitenumfang | 9 |

Fachzeitschrift | Mathematische Zeitschrift |

Jahrgang | 306 |

Publikationsstatus | Veröffentlicht - 24 Nov. 2023 |

## Abstract

A finite group G with center Z is of central type if there exists a fully ramified character λ∈ Irr (Z) , i. e. the induced character λ^{G} is a multiple of an irreducible character. Howlett–Isaacs have shown that G is solvable in this situation. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p≠ 5 . We show that there are no exceptions for p= 5 , i. e. every group of p-central type is solvable. Gagola proved that every solvable group can be embedded in G/Z for some group G of central type. We generalize this to groups of p-central type. As an application we construct some interesting non-nilpotent blocks with a unique Brauer character. This is related to a question by Kessar and Linckelmann.

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**Groups of p-central type.**/ Sambale, Benjamin.

in: Mathematische Zeitschrift, Jahrgang 306, 3, 24.11.2023.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

*Mathematische Zeitschrift*, Jg. 306, 3. https://doi.org/10.1007/s00209-023-03406-3

*Mathematische Zeitschrift*,

*306*, Artikel 3. https://doi.org/10.1007/s00209-023-03406-3

}

TY - JOUR

T1 - Groups of p-central type

AU - Sambale, Benjamin

N1 - Funding Information: The work on this paper started during a research stay at the University of Valencia in February 2023. I thank Gabriel Navarro and Alexander Moretó for the great hospitality received there. Theorem was initiated by a question of Britta Späth at the Oberwolfach workshop “Representations of finite groups” (ID 2316) in April 2023. I further thank Radha Kessar for some helpful discussions on this paper. This paper is supported by the German Research Foundation (SA 2864/4-1).

PY - 2023/11/24

Y1 - 2023/11/24

N2 - A finite group G with center Z is of central type if there exists a fully ramified character λ∈ Irr (Z) , i. e. the induced character λG is a multiple of an irreducible character. Howlett–Isaacs have shown that G is solvable in this situation. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p≠ 5 . We show that there are no exceptions for p= 5 , i. e. every group of p-central type is solvable. Gagola proved that every solvable group can be embedded in G/Z for some group G of central type. We generalize this to groups of p-central type. As an application we construct some interesting non-nilpotent blocks with a unique Brauer character. This is related to a question by Kessar and Linckelmann.

AB - A finite group G with center Z is of central type if there exists a fully ramified character λ∈ Irr (Z) , i. e. the induced character λG is a multiple of an irreducible character. Howlett–Isaacs have shown that G is solvable in this situation. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p≠ 5 . We show that there are no exceptions for p= 5 , i. e. every group of p-central type is solvable. Gagola proved that every solvable group can be embedded in G/Z for some group G of central type. We generalize this to groups of p-central type. As an application we construct some interesting non-nilpotent blocks with a unique Brauer character. This is related to a question by Kessar and Linckelmann.

KW - Fully ramified characters

KW - Groups of central type

KW - Howlett–Isaacs theorem

UR - http://www.scopus.com/inward/record.url?scp=85178088774&partnerID=8YFLogxK

U2 - 10.1007/s00209-023-03406-3

DO - 10.1007/s00209-023-03406-3

M3 - Article

AN - SCOPUS:85178088774

VL - 306

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

M1 - 3

ER -