A problem in the Kourovka notebook concerning the number of conjugacy classes of a finite group

TL;DR

This paper investigates a longstanding conjecture about the number of conjugacy classes in finite groups, providing new constraints on potential minimal counterexamples and advancing understanding of the problem.

Contribution

It establishes properties that a minimal soluble counterexample must have, including bounds on Fitting height and order, contributing to the ongoing study of the conjecture.

Findings

A minimal soluble counterexample must have Fitting height at least 3.

Such a counterexample must have order at least 2000.

The conjecture remains open despite these constraints.

Abstract

In this paper, we consider Problem 14.44 in the Kourovka notebook, which is a conjecture about the number of conjugacy classes of a finite group. While elementary, this conjecture is still open and appears to elude any straightforward proof, even in the soluble case. However, we do prove that a minimal soluble counterexample must have certain properties, in particular that it must have Fitting height at least 3 and order at least 2000.