Finite groups have more conjugacy classes

TL;DR

This paper establishes a lower bound on the number of conjugacy classes in finite groups, improving previous results and confirming a specific conjecture for groups with a trivial solvable radical.

Contribution

It proves a new lower bound on conjugacy classes in finite groups and confirms Bertram's conjecture for a subclass of groups.

Findings

Established a lower bound involving logarithms of group order

Improved upon earlier bounds by Pyber and Keller

Confirmed Bertram's conjecture for groups with trivial solvable radical

Abstract

We prove that for every $\epsilon > 0$ there exists a $\delta > 0$ so that every group of order $n \geq 3$ has at least $\delta \log_{2} n/{(\log_{2} \log_{2} n)}^{3+\epsilon}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order $n$ has more than $\log_{3}n$ conjugacy classes. We answer Bertram's question in the affirmative for groups with a trivial solvable radical.