pi-blocks of pi-separable groups. III (Q1801906)

[For part II see ibid. 124, No. 1, 236-269 (1989; Zbl 0684.20008).] Let \(\pi\) be a set of prime numbers. A \(\pi\)-block of a finite group \(G\) is a minimal nonempty set of characters of \(G\), which is a union of (Brauer) \(p\)-blocks for all \(p \in \pi\). The main result, which is used to prove a conjecture of T. Hawkes, is Theorem: Let \(G\) be a finite solvable group and let \(\sigma\) and \(\tau\) be sets of primes. If \(b_ \sigma\) and \(b_ \tau\) are \(\sigma\)- and \(\tau\)-blocks of \(G\) respectively, then \(b_ \sigma \cap b_ \tau \neq \emptyset\) if and only if \(b_ \sigma\) and \(b_ \tau\) lie in the same (\(\sigma \cup \tau\))- block of \(G\).