Definition:Internal Group Direct Product/General Definition/Definition by Unique Expression

Definition
Let $\sequence {H_n} = \struct {H_1, \circ {\restriction_{H_1} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ be a (finite) sequence of subgroups of a group $\struct {G, \circ}$

where $\circ {\restriction_{H_1} }, \ldots, \circ {\restriction_{H_n} }$ are the restrictions of $\circ$ to $H_1, \ldots, H_n$ respectively.

The group $\struct {G, \circ}$ is the internal group direct product of $\sequence {H_n}$ :


 * $(1): \quad$ Each $H_1, H_2, \ldots, H_n$ is a normal subgroup of $G$


 * $(2): \quad$ Each element $g$ of $G$ can be expressed uniquely in the form:
 * $g = h_1 \circ h_2 \circ \cdots \circ h_n$
 * where $h_i \in H_i$ for all $i \in \set {1, 2, \ldots, n}$.

Also see

 * Equivalence of Definitions of Internal Group Direct Product