Details
| Original language | English |
|---|---|
| Pages (from-to) | 103-115 |
| Number of pages | 13 |
| Journal | Advances in Group Theory and Applications |
| Volume | 8 |
| Publication status | Published - 2019 |
| Externally published | Yes |
Abstract
Slattery has generalized Brauer’s theory of p-blocks of finite groups to π-blocks of π-separable groups where π is a set of primes. In this setting we show that the order of a defect group of a π-block B is bounded in terms of the number of irreducible characters in B. This is a variant of Brauer’s Problem 21 and generalizes Külsham-mer’s corresponding theorem for p-blocks of p-solvable groups. At the same time, our result generalizes Landau’s classical theorem on the number of conjugacy classes of an arbitrary finite group. The proof relies on the classification of finite simple groups.
Keywords
- Brauer’s Problem 21, Number of characters, π-blocks
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Advances in Group Theory and Applications, Vol. 8, 2019, p. 103-115.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Landau’s theorem for π-blocks of π-separable groups
AU - Sambale, Benjamin
N1 - Funding information: * The author is supported by the German Research Foundation (SA 2864/1-1 and SA 2864/3-1)
PY - 2019
Y1 - 2019
N2 - Slattery has generalized Brauer’s theory of p-blocks of finite groups to π-blocks of π-separable groups where π is a set of primes. In this setting we show that the order of a defect group of a π-block B is bounded in terms of the number of irreducible characters in B. This is a variant of Brauer’s Problem 21 and generalizes Külsham-mer’s corresponding theorem for p-blocks of p-solvable groups. At the same time, our result generalizes Landau’s classical theorem on the number of conjugacy classes of an arbitrary finite group. The proof relies on the classification of finite simple groups.
AB - Slattery has generalized Brauer’s theory of p-blocks of finite groups to π-blocks of π-separable groups where π is a set of primes. In this setting we show that the order of a defect group of a π-block B is bounded in terms of the number of irreducible characters in B. This is a variant of Brauer’s Problem 21 and generalizes Külsham-mer’s corresponding theorem for p-blocks of p-solvable groups. At the same time, our result generalizes Landau’s classical theorem on the number of conjugacy classes of an arbitrary finite group. The proof relies on the classification of finite simple groups.
KW - Brauer’s Problem 21
KW - Number of characters
KW - π-blocks
UR - http://www.scopus.com/inward/record.url?scp=85078277910&partnerID=8YFLogxK
U2 - 10.32037/agta-2019-013
DO - 10.32037/agta-2019-013
M3 - Article
AN - SCOPUS:85078277910
VL - 8
SP - 103
EP - 115
JO - Advances in Group Theory and Applications
JF - Advances in Group Theory and Applications
SN - 2499-1287
ER -