Definition:Group Direct Product/General Definition

Definition
Let $\left\{ \left({G_i, \circ_i}\right)\right\}_{i \in I}$ be a family of groups.

Let $\displaystyle G = \prod_{i\in I} G_i$ be their cartesian product.

The operation induced on $G$ by $\left(\circ_i\right)_{i\in I}$ is the operation $\circ$ defined by:


 * $\left\langle g_i\right\rangle_{i \mathop \in I}\circ \left\langle h_i\right\rangle_{i \mathop \in I} = \left\langle g_i \circ_i h_i\right\rangle_{i \mathop \in I}$

In External Direct Product of Groups is Group, it is shown that this gives $\left({G, \circ}\right)$ the structure of a group.

The group $\left({G, \circ}\right)$ is called the (external) direct product of $\left\{ \left({G_i, \circ_i}\right)\right\}_{i \in I}$.

Finite Direct Product
Let $\left({G_1, \circ_1}\right), \left({G_2, \circ_2}\right), \ldots, \left({G_n, \circ_n}\right)$ be groups.

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

The operation induced on $G$ by $\circ_1, \ldots, \circ_n$ is the operation $\circ$ defined by:


 * $\left({g_1, g_2, \ldots, g_n}\right) \circ \left({h_1, h_2, \ldots, h_n}\right) = \left({g_1 \circ_1 h_1, g_2 \circ_2 h_2, \ldots, g_n \circ_n h_n}\right)$

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

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