On the number of irreducible characters in a 2-block

Landrock, Peter

doi:10.1016/0021-8693(81)90272-6

Cited by 26 publications

(29 citation statements)

References 11 publications

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“…Now a similar analysis of the contributions reveals that Q has the form (c) (see above) and k 0 (B) = 8. Again (5G) in [4] implies k 1 (B) = 6 (this follows also from Corollary 1.4 in [17]). Since the subsections (x, b x ) and (x −1 , b x ) are conjugate in G, the generalized decomposition numbers d x χϕ x and d xz χϕ xz are integral.…”

Section: -Blocks With Defectmentioning

confidence: 61%

Cartan matrices and Brauerʼs k(B)-conjecture II

Sambale

2011

Journal of Algebra

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This paper continues Sambale (2011) [28]. We show that the methods developed there also work for odd primes. In particular we prove Brauer's k(B)-conjecture for defect groups which contain a central, cyclic subgroup of index at most 9. As a consequence, the k(B)-conjecture holds for 3-blocks of defect at most 3. In the second part of the paper we illustrate the limits of our methods by considering an example. Then we use the work of Kessar, Koshitani and Linckelmann [13] (and thus the classification) to show that the k(B)-conjecture is satisfied for 2-blocks of defect 5 except for the extraspecial defect group D 8 * D 8 . As a byproduct we also obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups. Some proofs rely on computer computations with GAP (The GAP Group, 2008 [10]).

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Section: -Blocks With Defectmentioning

confidence: 61%

Cartan matrices and Brauerʼs k(B)-conjecture II

Sambale

2011

Journal of Algebra

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“…Obviously, the smallest non-metacyclic group C 3 2 comes to mind. Landrock [176] gave partial results here, and later the case was settled by Kessar, Koshitani and Linckelmann [148] using the classification of the finite simple groups. The result also follows easily from Theorem 7.14 (still using the classification).…”

Section: Results On the Kb/-conjecturementioning

confidence: 85%

Blocks of Finite Groups and Their Invariants

Sambale

2014

Lecture Notes in Mathematics

155

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“…Proof. If b is a principal block then r = 3 and G = J 1 , hence the result follows from [48,Theorem 3.8]. Suppose that b is a non-principal block; by Proposition 6.1 we may assume that b is faithful.…”

Section: Sporadic Finite Simple Groupsmentioning

confidence: 95%

“…There is always at least one character in Irr K (G, b) having height zero, and it has been conjectured by Brauer that P is abelian if and only if all characters in Irr K (G, b) have height zero. The following summary of some of Landrock's results in [48]…”

Section: Reduction Techniquesmentioning

confidence: 99%