A lower bound for the number of conjugacy classes of finite groups

TL;DR

This paper extends a known lower bound on the number of conjugacy classes from solvable groups to all finite groups for large primes, providing a broader understanding of group structure.

Contribution

It generalizes a previous result by proving the lower bound holds for all finite groups when the prime is sufficiently large.

Findings

The lower bound of 2√(p-1) conjugacy classes applies to all finite groups for large primes p.

The result broadens the scope from solvable groups to arbitrary finite groups.

Supports the conjecture that the bound is asymptotically tight for large primes.

Abstract

In 2000, L. H\'{e}thelyi and B. K\"{u}lshammer proved that if $p$ is a prime number dividing the order of a finite solvable group $G$, then $G$ has at least $2\sqrt{p-1}$ conjugacy classes. In this paper we show that if $p$ is large, the result remains true for arbitrary finite groups.