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

TL;DR

This paper explores how the number of p-regular and p-singular conjugacy classes in finite groups influences their structure, extending Landau's theorem and revealing strong restrictions under certain conditions.

Contribution

It establishes new bounds and structural restrictions for finite groups based on p-regular and p-singular classes, generalizing Landau's theorem for these cases.

Findings

Finite groups with trivial O_p(G) have bounded |G/F(G)|_{p'} based on p-regular classes.

Structural restrictions are derived for groups with specific conjugacy class counts.

Finiteness results depend on unresolved number-theoretic problems like Fermat primes.

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).