Definition:Independent Subgroups/Definition 1

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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 if and only if:

$\ds \prod_{k \mathop = 1}^n h_k = e \iff \forall k \in \set {1, 2, \ldots, n}: h_k = e$

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 if and only if all of those elements are the identity.

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