On \(\pi\)-subpairs and the third main theorem for \(\pi\)-blocks (Q1825271)

Let G be a finite group, and let \(\pi\) be a set of primes. The authors are interested in the theory of \(\pi\)-blocks of irreducible characters of G. Generalizing the Alperin-Broué approach for a single prime p, they define a \(\pi\)-subpair of G to be a pair (P,B) where P is a nilpotent \(\pi\)-subgroup of G and B is a \(\pi\)-block of \(C_ G(P)\). They then use the Second Main Theorem for \(\pi\)-blocks to define an inclusion between such \(\pi\)-subpairs. It turns out that a substantial part of the theory for a single prime carries over to a set of primes. However, the maximal \(\pi\)-subpairs containing a given \(\pi\)-subpair (1,A) need not be conjugate in G. Moreover, there is no analogue of the Third Main Theorem in general. Nevertheless, the authors are able to prove a version of the Third Main Theorem in case G contains a cyclic Hall \(\pi\)-subgroup. This result makes use of E. C. Dade's theory of p-blocks with cyclic defect groups.