Hall \(\pi\)-subgroups and conjugacy classes (Q915854)

The authors concern themselves with problems of counting conjugacy classes of elements and of subgroups of a given group. Normally for the groups in question there is some condition on their \(\pi\)-structure for some sets of primes \(\pi\). There is a considerable amount of notation which makes it difficult to paraphrase concisely their results. However in order to give the flavour of the results let me quote one: Theorem 2.4: Let G be a finite group containing a nilpotent Hall \(\pi\)-subgroup. Let K be a \(\pi\)-subgroup of G. Then the number of Hall \(\pi\)-subgroups of G containing K is congruent to 1 modulo the greatest common divisor of the set \(\{\) p-1\(|\) where \(p\not\in \pi\) but p divides \(| G| \}\). The proof of the theorem uses quite delicate counting arguments but does not depend on much machinery.