Landau's theorem, fields of values for characters, and solvable groups

TL;DR

This paper generalizes Landau's theorem for solvable groups by relating the number of conjugacy classes of prime power order elements to irreducible characters over fields extended by prime power roots of unity, providing new bounds.

Contribution

It proves a new inequality linking conjugacy classes and irreducible characters for solvable groups and extends Landau's theorem using field extension techniques.

Findings

Bound on conjugacy classes of prime power order elements

Bound on group order in terms of irreducible characters

Generalization of Landau's theorem for solvable groups

Abstract

When $G$ is solvable group, we prove that the number of conjugacy classes of elements of prime power order is less than or equal to the number of irreducible characters with values in fields where $\mathbb {Q}$ is extended by prime power roots of unity. We then combine this result with a theorem of H\'ethelyi and K\"ulshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order to bound the order of a solvable group by the number of irreducible characters with values in fields extended by prime power roots of unity. This yields for solvable groups a generalization of Landau's theorem.