Internal Direct Product Theorem/General Result

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

Let $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of subgroups of $G$.

Then $G$ is the internal group direct product of $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ :


 * $(1): \quad G = H_1 H_2 \cdots H_n$
 * $(2): \quad \sequence {H_k}_{1 \mathop \le k \mathop \le n}$ is a sequence of independent subgroups
 * $(3): \quad \forall k \in \set {1, 2, \ldots, n}: H_k \lhd G$

where $H_k \lhd G$ denotes that $H_k$ is a normal subgroup of $G$.