\begin{definition}[Definition:Independent Subgroups/Definition 1]
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}}:
:$\ds \prod_{k \mathop = 1}^n h_k = e \iff \forall k \in \set {1, 2, \ldots, n}: h_k = e$
where $h_k \in H_k$ for all $k \in \set {1, 2, \ldots, n}$.
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}