Group theory and generalizations Definition

Direct product of groups

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Definition

Let $G$ and $H$ be groups. The direct product $G \times H$ is the set of ordered pairs $(g, h)$ with $g \in G$ and $h \in H$ , with the operation $(g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)$ .

Remarks

The identity is $(e_G, e_H)$ , and the inverse of $(g, h)$ is $(g^{-1}, h^{-1})$ . If $G$ and $H$ are finite, then $|G \times H| = |G| \cdot |H|$ . The order of $(g, h)$ is the least common multiple of the orders of $g$ and $h$ , when both are finite. For example, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is cyclic of order $6$ , but $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ is not cyclic.

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Group concept
Direct product of groups definition

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Tag	`0YP7WYXF9V`
URI	`m2://0YP7WYXF9V`
Aliases	`external-direct-product`
Kind	definition
Writing	W1 stated
MSC	20E34
Defines	Direct product of groups
Uses	Group

Direct product of groups

Definition

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