Benjamin Sambale

Blocks of finite groups and their invariants

Zusammenfassung.- Introduction.- Fundamentals.- General results and methods.- Applications.- Bibliopraphy.- Index.- List of tables.

On Loewy lengths of blocks

We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. Afterwards we discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p = 2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.

2-Blocks with minimal nonabelian defect groups II

We determine the structure of 2-blocks with minimal nonabelian defect groups, by making use of the classification of finite simple groups.

The 2-blocks of defect 4

We show that the major counting conjectures of modular representation theory are satisfied for 2-blocks of defect at most 4 except one possible case. In particular we determine the invariants of such blocks.

Blocks with central product defect group _{2ⁿ}∗_{2^{}}

We determine the numerical invariants of blocks with defect group D2n C2m = Q2n C2m (central product), where n 3 and m 2. As a consequence, we prove Brauer’s k(B)-conjecture, Olsson’s conjecture (and more generally Eaton’s conjecture), Brauer’s height zero conjecture, the Alperin-McKay conjecture, Alperin’s weight conjecture and Robinson’s ordinary weight conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper continues (Sambale, 2012, [21]).

Fusion Systems on Bicyclic 2-Groups

We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups (see [Craven-Glesser, 2012] and [Sambale, 2012]). As an application we prove Olssons Conjecture for all blocks with bicyclic defect groups.

Cartan matrices and Brauer’s k(B)-conjecture III

For a block B of a finite group we prove that

Broué's isotypy conjecture for the sporadic groups and their covers and automorphism groups

{k(B) \le ({\rm det} C - 1)/l(B)+l(B) \le {\rm det} C}