Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

Abstract

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O p (G) = 1 then |G/F(G)| p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and k p-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).