Definition:Independent Subgroups

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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:

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

where $\left[{m \,.\,.\, n}\right]$ is to be interpreted as the (closed) integer interval from $m$ to $n$.

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

From Condition for Subgroups to be Independent we have that $H_1, H_2, \ldots, H_n$ are independent iff:

$\displaystyle \forall k \in \left[{2 \,.\,.\, n}\right]: \left({\prod_{j=1}^{k-1} H_j}\right) \cap H_k = \left\{{e}\right\}$