Elliptic pair of subgroups

This article defines a symmetric relation on the collection of subgroups inside the same group.

Definition

Definition with symbols

Let $H$ and $K$ be subgroups of a group $G$. We say that $(H,K)$ form an elliptic pair of subgroups if there exists a positive integer $n$ such that:

$<H,K>=(HK{)}^n$

In other words, every element that can be expressed as a product of elements from $H$ and $K$, can be expressed as a product of length at most $2n$ (with alternating elements from $H$ and $K$).