Definition:Independent Subgroups

Definition
Let $$G$$ be a group whose identity is $$e$$.

Let $$\left \langle {H_n} \right \rangle$$ be a sequence of subgroups of $$G$$.

Let $$h_k \in H_k$$ for all $$k \in \left[{1 \,. \, . \, n}\right]$$.

The subgroups $$H_1, H_2, \ldots, H_n$$ are described as independent iff:


 * $$\prod_{k=1}^n h_k = e \iff \forall k \in \left[{1 \, . \, . \, n}\right]: h_k = e$$

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

Also see
From Condition for Subgroups to be Independent we have that $$H_1, H_2, \ldots, H_n$$ are independent iff:
 * $$\forall k \in \left[{2 \, . \, . \, n}\right]: \left({\prod_{j=1}^{k-1} H_j}\right) \cap H_k = \left\{{e}\right\}$$