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Further evidence for conjectures in block theory. (English) Zbl 1290.20010
This paper studies some well-known conjectures in block theory of finite groups and proves some of them in special cases, often generalizing earlier results. For example, by a 1981 result of Olsson Brauer’s \(k(B)\)-conjecture holds if the number of Brauer characters in the block \(B\) is 1 or 2. The author obtains the same conclusion in case that there is a major subsection \((z,b_z)\) with \(l(b_z)\leq 2\). Furthermore, Brauer’s \(k(B)\)-conjecture is proved for 2-blocks of defect \(\leq 5\), and also for defect groups with a cyclic subgroup of index at most 4. Alperin’s weight conjecture and other conjectures are verified for several infinite families of nonnilpotent blocks; Olsson’s conjecture is also studied from various angles.

MSC:
20C20 Modular representations and characters
20C15 Ordinary representations and characters
Software:
GAP
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