Definition:Independent Subgroups/Definition 1

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 :


 * $\displaystyle \prod_{k \mathop = 1}^n h_k = e \iff \forall k \in \left\{{1, 2, \ldots, n}\right\}: h_k = e$

where $h_k \in H_k$ for all $k \in \left\{{1, 2, \ldots, n}\right\}$.

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