# Square-free class sizes in products of groups

## TL;DR

This paper investigates the structure of factorized groups, particularly when conjugacy class sizes of certain elements are not divisible by the square of a prime, with special focus on mutually permutable products.

## Contribution

It provides new structural insights into groups factorized as $G=AB$, especially under conditions on conjugacy class sizes related to a prime $p$.

## Key findings

- Structural properties of $G=AB$ when conjugacy class sizes are not divisible by $p^2$
- Special results for mutually permutable products
- Conditions influencing the group's composition based on class size divisibility

## Abstract

We obtain some structural properties of a factorised group $G = AB$, given that the conjugacy class sizes of certain elements in $A\cup B$ are not divisible by $p^2$, for some prime $p$. The case when $G = AB$ is a mutually permutable product is especially considered.