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 $G$.

Definition 1

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 $k \in \set {1, 2, \ldots, n}$.

Definition 2

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

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

Also see

• Equivalence of Definitions of Independent Subgroups
• Conditions for Internal Group Direct Product