Wreath products and the non-coprime $k(GV)$ problem

TL;DR

This paper proves a bound on the number of conjugacy classes in certain wreath products, solving a case of the non-coprime $k(GV)$ problem and answering a question for semiprimitive groups without relying on the classification of finite simple groups.

Contribution

It establishes a new bound on $k(G)$ for wreath products with semiprimitive groups, advancing understanding of the non-coprime $k(GV)$ problem.

Findings

Proves $k(G) \\leq k^n$ for wreath products with semiprimitive $H$.

Provides an affirmative answer to Garzoni and Gill's question.

Avoids use of the classification of finite simple groups.

Abstract

Let $G = X \wr H$ be the wreath product of a nontrivial finite group $X$ with $k$ conjugacy classes and a transitive permutation group $H$ of degree $n$ acting on the set of $n$ direct factors of $X^n$. If $H$ is semiprimitive, then $k(G) \leq k^n$ for every sufficiently large $n$ or $k$. This result solves a case of the non-coprime $k(GV)$ problem and provides an affirmative answer to a question of Garzoni and Gill for semiprimitive permutation groups. The proof does not require the classification of finite simple groups.