Cartesian Product
The cartesian product of two sets A and B is the set of all ordered pairs (a,b) with a and b elements of A and B respectively. This new set, the cartesian product, is usually denoted A × B. Formally:
A × B : = {(a,b) | a ∈ A, b ∈ B}.
This definition may be extended in the obvious way to form the cartesian product of more than two sets.
~ Group
Given two groups G1 and G2, with corresponding operations +1 and +2, we can define the cartesian product product group G = G1 × G2. Its elements are those of the cartesian product of the sets of elements of the two original groups. The group operation +, is defined by:
(x1, x2) + (y1, y2) : = (x1 +1 y1 , x2 +2 y2).
It can be easily varified that G is indeed a group.