On the number of conjugacy classes of a primitive permutation group with nonabelian socle

TL;DR

This paper investigates the number of conjugacy classes in primitive permutation groups with nonabelian socles, establishing bounds and characterizing specific families of such groups.

Contribution

It proves bounds on the number of conjugacy classes in these groups and identifies explicit families where the bounds are attained.

Findings

If $k(G)<n/2$, then $k(G)=o(n)$ as $n$ grows large.

The paper characterizes specific families of primitive groups with nonabelian socles.

Provides bounds and classifications for conjugacy class counts in these groups.

Abstract

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$. We prove that either $k(G)<n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$, or $G$ belongs to explicit families of examples.