Let \(G\) be a finite group, \(p\) a prime, and \((K,R,k)\) a \(p\)-modular system with \(K\) and \(k\) sufficiently large. Let \(\nu\) be the valuation on \(K\), normalized so that \(\nu(p)=1\). For an \(RG\) lattice \(U\) lying in a block \(B\) of \(G\), the height of \(U\) is defined as \(\nu(\text{rank}_RU)-\nu(|G|)+d(B)\), where \(d(B)\) is the defect of \(B\). Heights of \(kG\)-modules are defined similarly. Here, modules of height zero are considered.
Motivated by work of Broué, the author first generalizes the notion of block linkage, and finds its basic properties. He then gives a criterion for a block to be induced from a normal subgroup. Next, the indecomposable modules of height zero in a block \(B\) with defect group \(D\) are characterized as those that have vertex conjugate to \(D\) and source of rank prime to \(p\). As an application, it is shown that if \(\chi\) is a character of height zero and \(N\) is a normal subgroup of \(G\), then the restriction \(\chi_N\) has an irreducible constituent of height zero, and the characters of \(N\) that so arise are determined.
A generalization of the Isaacs-Smith characterization of groups of \(p\)- length one is obtained. Finally, Brauer conjectured that blocks with abelian defect groups contain only characters of height zero. Berger and Knörr showed that it suffices to prove this for quasi-simple groups; the author gives a new proof of their result.