On the number of irreducible characters in a 2-block

Abstract

Both of these conjectures have been verified for D cyclic by Dade [9] or 1 D 1 <pz by Brauer and Feit [7] (for proof see Brauer [3, II]). In addition, the first conjecture has been verified if the inertial index of the block is 1 (see Brauer [4, IV]), while the second conjecture has been partly verified for psolvable groups by Fong. In one of several discussions I had with R. Brauer in 1974, he suggested to find a verification of these conjectures in the case of a principal 2-block with H, @ Z, @ Z, as defect group without involving the classification of groups with this Sylow 2-subgroup, from which the conjectures may quite easily be checked to hold. In [ 181, Ward was forced to do this for groups of Ree-type anyway, but the computations were quite involved, and the very special properties of these groups, including their order, were used heavily. In Theorem 2 of Section 2, we prove that the first conjecture holds in general for any defect group of order (less than or) equal to 8, while Theorem 8 of Section 3 states that the second conjecture hold if the block in addition is assumed to be principal. We shall not depend in our work on the classification of these groups. In particular we give an alternate and shorter proof of the result in [ 181 mentioned above without using any other property