Bounds on the number of conjugacy classes of the symmetric and alternating groups

TL;DR

This paper proves that Pyber's inequality, relating the number of conjugacy classes of a finite group to those of its Sylow subgroups, holds specifically for symmetric and alternating groups, using computational tools.

Contribution

The paper establishes Pyber's inequality for symmetric and alternating groups, providing a significant case verification using computational methods.

Findings

Pyber's inequality holds for symmetric groups

Pyber's inequality holds for alternating groups

Computational verification with GAP confirms the results

Abstract

Let $G$ be a finite group with Sylow subgroups $P_1,\ldots,P_n$, and let $k(G)$ denote the number of conjugacy classes of $G$. Pyber asked if $k(G) \leq \prod_{i=1}^n k(P_i)$ for all finite groups $G$. With the help of GAP, we prove that Pyber's inequality holds for all symmetric and alternating groups.