Zeros of irreducible characters in factorised groups

TL;DR

This paper investigates the structural properties of factorised finite groups, focusing on the behavior of prime power order elements that are zeros or non-zeros of irreducible characters, revealing new insights into their group structure.

Contribution

It provides new structural results for factorised groups based on conditions on prime power order elements that are zeros or non-zeros of irreducible characters.

Findings

Characterizes zero elements in factorised groups.

Establishes conditions linking element vanishing and group structure.

Provides criteria for prime power order elements in factorised groups.

Abstract

An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there exists an irreducible character $\chi$ of $G$ such that $\chi(g)=0$; in this case, $g$ is also called a zero of $G$. The aim of this paper is to obtain structural properties of a factorised group $G=AB$ when we impose some conditions on prime power order elements $g\in A\cup B$ which are (non-)vanishing in $G$.