\begin{definition}[Definition:Independent Subgroups/Definition 2]
Let $G$ be a group whose identity is $e$.
Let $\sequence {H_n}$ be a sequence of subgroups of $G$.
The subgroups $H_1, H_2, \ldots, H_n$ are '''independent''' {{iff}}:
:$\ds \forall k \in \set {2, 3, \ldots, n}: \paren {\prod_{j \mathop = 1}^{k - 1} H_j} \cap H_k = \set e$
That is, the product of any elements from different $H_k$ instances forms the identity {{iff}} ''all'' of those elements ''are'' the identity.
\end{definition}