Definition:Independent Subgroups/Definition 2

Definition
Let $G$ be a group whose identity is $e$.

Let $\left \langle {H_n} \right \rangle$ be a sequence of subgroups of $G$.

The subgroups $H_1, H_2, \ldots, H_n$ are independent :


 * $\displaystyle \forall k \in \left\{{2, 3, \ldots, n}\right\}: \left({\prod_{j \mathop = 1}^{k-1} H_j}\right) \cap H_k = \left\{{e}\right\}$

That is, the product of any elements from different $H_k$ instances forms the identity all of those elements are the identity.

Also see

 * Equivalence of Definitions of Independent Subgroups