On the Vertices of Irreducible Modules

Abstract

The same result holds if D is a vertex of an irreducible FG-module Me B. This second assertion generalizes some known results: For p-solvable groups G, it has been shown by Hamernik and Michler [10], that for any irreducible module M in a block B of FG, a vertex of M contains-up to conjugation-the center of a defect group of B. In [13], Michler has shown that all irreducible modules in a block with cyclic defect group A have A as a vertex. A further result in this context was given by Landrock and Michler ([11], Theorem 3.7) as a by-product of their analysis of the principal 2-blocks of the smallest Janko group and the groups of Ree-type. The proof here uses properties of the Green functor to reduce to the case D < G. Then one has to study direct summands of a module induced from a normal subgroup. This is done by inspection of the ring of endomorphisms (Sections 1 and 2). Using these results, the existence of certain B-pairs (see Definition 0.3 below) is shown in Section 3. The results mentioned at the beginning follow then from work done by Brauer and Olsson. The techniques used in the first sections yield a criterion for an ordinary irreducible character X to be of height 0: If F is algebraically closed and M an R-module affording X, then the character X belonging to the p-block B is of height 0 if and only if a vertex of M is a defect group of B and p does not divide the R-rank of a source of M. The proof is given in Section 4.