Conjugacy class numbers and \(\pi \)-subgroups (Q2057574)

The number \(l(G)\) of \(p\)-regular classes of a finite group \(G\) is a key invariant in modular representation theory. The main question is wether \(l(G)\) can be bounded by the number of conjugacy classes of some subgroup of \(G\) of order not divisible by \(p\). This would have consequences for the Malle-Robinson \(l(B)\)-conjecture. In this paper, the authors investigate a \(\pi\)-version of this, for sets of primes \(\pi\). As part of investigations, the authors study finite groups that have more conjugacy classes than any of their proper subgroups. These groups naturally appear in questions on bounding from above the number of conjugacy classes of a group, and were considered by G. R. Robinson and J. G. Thompson in the context of the \(k(GV)\)-problem.