Definition:Independent Subgroups/Definition 1

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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 $G$.


The subgroups $H_1, H_2, \ldots, H_n$ are independent if and only if:

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

where $h_k \in H_k$ for all $k \in \left\{{1, 2, \ldots, n}\right\}$.


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.


Also see


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