Elliptic pair of subgroups

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:

$$ = (HK)^n$$

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 $$K$$).

Stronger relations

 * Permuting subgroups