# Definition:Independent Subgroups/Definition 2

## Definition

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:

$\displaystyle \forall k \in \set {2, 3, \ldots, n}: \paren {\prod_{j \mathop = 1}^{k - 1} H_j} \cap H_k = \set e$

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