Conjugacy classes, characters and products of elements

TL;DR

This paper explores properties of finite groups related to conjugacy classes and characters, extending a recent result on nilpotency characterized by element orders, and employs classification of finite simple groups in its proofs.

Contribution

It investigates groups with specific conjugacy class and character multiplication properties, providing new characterizations and weakening existing hypotheses.

Findings

Groups with $(xy)^G=x^Gy^G$ have particular structural properties.

Character multiplicativity $ ext{chi}(xy)= ext{chi}(x) ext{chi}(y)$ characterizes certain group classes.

Some results rely on the classification of finite simple groups.

Abstract

Recently, Baumslag and Wiegold proved that a finite group $G$ is nilpotent if and only if $o(xy)=o(x)o(y)$ for every $x,y\in G$ of coprime order. Motivated by this result, we study the groups with the property that $(xy)^G=x^Gy^G$ and those with the property that $\chi(xy)=\chi(x)\chi(y)$ for every complex irreducible character $\chi$ of $G$ and every nontrivial $x, y \in G$ of pairwise coprime order. We also consider several ways of weakening the hypothesis on $x$ and $y$. While the result of Baumslag and Wiegold is completely elementary, some of our arguments here depend on (parts of) the classification of finite simple groups.