Prime divisors and the number of conjugacy classes of finite groups

TL;DR

This paper establishes a lower bound on the number of conjugacy classes in finite groups based on prime divisors of the index of their Fitting subgroup, proposing a universal constant and conjecturing an optimal value.

Contribution

The authors prove a universal lower bound involving prime divisors for the number of conjugacy classes and conjecture an improved bound for solvable groups.

Findings

Existence of a universal constant D for the bound

Lower bound of conjugacy classes proportional to p/log p

Conjecture that D can be taken as 1 for all groups

Abstract

We prove that there exists a universal constant $D$ such that if $p$ is a prime divisor of the index of the Fitting subgroup of a finite group $G$, then the number of conjugacy classes of G is at least $Dp/log_2 p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$.