On the number of conjugacy classes of a permutation group

TL;DR

This paper establishes an upper bound on the number of conjugacy classes in permutation groups of degree at least 4, showing it does not exceed 5^{(n-1)/3}.

Contribution

The paper provides a new exponential upper bound on conjugacy classes for permutation groups of degree n.

Findings

Permutation groups of degree n ≥ 4 have at most 5^{(n-1)/3} conjugacy classes.

The bound improves understanding of the structure of permutation groups.

The result applies to all permutation groups of the specified degree.

Abstract

We prove that any permutation group of degree $n \geq 4$ has at most $5^{(n-1)/3}$ conjugacy classes.