Details
| Original language | English |
|---|---|
| Article number | 3 |
| Number of pages | 9 |
| Journal | Mathematische Zeitschrift |
| Volume | 306 |
| Publication status | Published - 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 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.
Keywords
- Fully ramified characters, Groups of central type, Howlett–Isaacs theorem
ASJC Scopus subject areas
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In: Mathematische Zeitschrift, Vol. 306, 3, 24.11.2023.
Research output: Contribution to journal › Article › Research › peer review
}
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 -