Direct product of groups

Given two groups $(A, \times_G)$ and $(B, \times_B)$, let us define a third structure $(X, \times_X)$ as
$X := A \times B$ (cartesian product)
$(a_1, b_1) \times_X (a_2, b_2) := (a_1 \times_A a_2, b_1 \times_B b_2)$

this structure $X$ is called the direct product of $A$ and $B$ and is denoted $A \otimes B$. The resulting structure $X$ is a group with identity $e_X = (e_A, e_B)$.
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