Cartan matrices and Brauerʼs k(B)-conjecture II

Abstract

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]).

Further evidence for conjectures in block theory

“…Now l(B) ≤ 3 follows easily. For the remaining cases, the claim was shown in [Sambale 2012c;Eaton et al 2012;Sambale 2011c;2012a;2013b;2013a] and the present paper.…”

“…In [Sambale 2011c], we verified Brauer's k(B)-conjecture for defect groups of order at most 32 but not isomorphic to the extraspecial group D 8 * D 8 . We are finally able to handle this remaining group as well.…”

“…Let B be a p-block with defect group D of the stated form. By [Sambale 2011c, Theorems 1 and 3], we may assume that there is a major B-subsection (z, b z ) such that D/ z ∼ = C 4 2 (in particular, p = 2) and B has inertial index 9. We apply [Feit 1982, Theorem V.9.17].…”