Definition:Group Direct Product/Finite Product

Definition

Let $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be groups.

Let $\ds G = \prod_{k \mathop = 1}^n G_k$ be their cartesian product.

Let $\circ$ be the operation defined on $G$ as:

$\circ := \tuple {g_1, g_2, \ldots, g_n} \circ \tuple {h_1, h_2, \ldots, h_n} = \tuple {g_1 \circ_1 h_1, g_2 \circ_2 h_2, \ldots, g_n \circ_n h_n}$

for all ordered $n$-tuples in $G$.

The group $\struct {G, \circ}$ is called the (external) direct product of $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$.

Also see

• External Direct Product of Groups is Group/Finite Product, where it is proved that $\struct {G, \circ}$ is a group.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \nu$