Direct product of groups
Given two groups (A,×G)(A, \times_G) and (B,×B)(B, \times_B), let us define a third structure (X,×X)(X, \times_X) as
X:=A×BX := A \times B (cartesian product) (a1,b1)×X(a2,b2):=(a1×Aa2,b1×Bb2)(a_1, b_1) \times_X (a_2, b_2) := (a_1 \times_A a_2, b_1 \times_B b_2)
this structure XX is called the direct product of AA and BB and is denoted ABA \otimes B. The resulting structure XX is a group with identity eX=(eA,eB)e_X = (e_A, e_B).

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