Definition:Group Direct Product/General Definition

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Let $\family {\struct {G_i, \circ_i} }_{i \mathop \in I}$ be a family of groups.

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

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

$\circ := \family {g_i}_{i \mathop \in I} \circ \family {h_i}_{i \mathop \in I} = \family {g_i \circ_i h_i}_{i \mathop \in I}$

for all sequences in $G$.

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