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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
GAP
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