# Prime power indices in factorised groups

## TL;DR

This paper investigates the structure of groups formed by the product of two subgroups, focusing on elements of prime power order and their indices, extending known results in group theory.

## Contribution

It characterizes the structure of factorized groups where prime power order elements have prime power indices, generalizing previous results for the case when the group equals one of its subgroups.

## Key findings

- Identifies structural properties of groups with prime power index elements
- Extends known results to broader classes of elements and group factorizations
- Provides conditions under which the group structure is constrained by element indices

## Abstract

Let the group $G = AB$ be the product of the subgroups $A$ and $B$. We determine some structural properties of $G$ when the $p$-elements in $A\cup B$ have prime power indices in $G$, for some prime $p$. More generally, we also consider the case that all prime power order elements in $A\cup B$ have prime power indices in $G$. In particular, when $G = A = B$ we obtain as a consequence some known results.