# 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.

## Key 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 $G$.