Homomorphisms from a finite group into wreath products

TL;DR

This paper studies the distribution of homomorphisms from finite groups into wreath products, showing that a natural induced distribution is nearly uniform and applying this to prove a conjecture about homomorphisms into Weyl groups.

Contribution

It demonstrates that the induced homomorphism distribution from random homomorphisms into wreath products is nearly uniform, and proves a conjecture on homomorphisms into Weyl groups of type D_n.

Findings

Induced distributions are close to uniform

Proved a conjecture of T. Müller on Weyl groups

Established properties of homomorphisms into wreath products

Abstract

Let $G$ be a finite group, $A$ a finite abelian group. Each homomorphism $\phi:G\to A\wr S_n$ induces a homomorphism $\bar{\phi}:G\to A$ in a natural way. We show that as $\phi$ is chosen randomly, then the distribution of $\bar{\phi}$ is close to uniform. As application we prove a conjecture of T. M\"uller on the number of homomorphisms from a finite group into Weyl groups of type $D_n$.