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 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.