A note on Olssonʼs Conjecture

For a p-block B of a finite group G with defect group D Olsson conjectured that k0(B)⩽|D:D′|, where k0(B) is the number of characters in B of height 0 and D′ denotes the commutator subgroup of D. Brauer deduced Olssonʼs Conjecture in the case where D is a dihedral 2-group using the fact that certain...

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Bibliographic Details
Published in:Journal of algebra 2014-01, Vol.398, p.364-385
Main Authors: Héthelyi, Lászlo, Külshammer, Burkhard, Sambale, Benjamin
Format: Article
Language:English
Subjects:
Block
Characters
Defect group
Height
p-Rank
Subsection
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Summary:For a p-block B of a finite group G with defect group D Olsson conjectured that k0(B)⩽|D:D′|, where k0(B) is the number of characters in B of height 0 and D′ denotes the commutator subgroup of D. Brauer deduced Olssonʼs Conjecture in the case where D is a dihedral 2-group using the fact that certain algebraically conjugate subsections are also conjugate in G. We generalize Brauerʼs argument for arbitrary primes p and arbitrary defect groups. This extends two results by Robinson. For p>3 we show that Olssonʼs Conjecture is satisfied for defect groups of p-rank 2 and for minimal non-abelian defect groups.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2012.08.010