# Products of groups and class sizes of $\pi$-elements

## TL;DR

This paper establishes structural criteria for finite factorised groups based on the conjugacy class sizes of certain $ ext{ extpi}$-elements, extending previous results for group products.

## Contribution

It provides new criteria for understanding the structure of factorised groups using conjugacy class sizes of $ ext{ extpi}$-elements, generalizing earlier findings.

## Key findings

- Criteria for finite factorised groups based on conjugacy class sizes.
- Extension of previous results to products of groups.
- Conditions involving $ ext{ extpi}$-numbers and $ ext{ extpi'}$-numbers.

## Abstract

We provide structural criteria for some finite factorised groups $G = AB$ when the conjugacy class sizes in $G$ of certain $\pi$-elements in $A\cup B$ are either $\pi$-numbers or $\pi'$-numbers, for a set of primes $\pi$. In particular, we extend for products of groups some earlier results.