Direct product of groups

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.

Explanations

L1 human

The direct product runs two groups side by side. An element is a pair, and you multiply the first parts in $G$ and the second parts in $H$. The two parts never interact.