We can construct a PascGalois Triangle -- the next twist on Pascal's Triangle -- by the following steps:

- Associate with every element in a finite group a different color.
- Place one element of the group down one side of the triangle.
- Place another (possibly different) group element down the the other side of the triangle.
- Using the group operation (generally referred to as "group multiplication"), generate the interior.

Pascal's triangle mod *n* for any integer *n* is one example of PascGalois Triangles, since the integers {0, 1, 2, ..., *n * - 1} under addition mod *n* are a finite group. We generally give names to groups so that we can easily refer to them. The set {0, 1, 2, ..., *n*-1} using addition mod *n* is called *Z _{n}*. This group is also called the

Here is another example: Consider the set of ordered pairs {(0,0), (1,0), (0,1), (1,1)} and the operation of combining two members of the set by adding the first components mod 2 to get the first component of the result and adding the second components mod 2 to get the second component of the result. Here is the multiplication table for this operation:

* | (0,0) | (0,1) | (1,0) | (1,1) |

(0,0) | (0,0) | (0,1) | (1,0) | (1,1) |

(0,1) | (0,1) | (0,0) | (1,1) | (1,0) |

(1,0) | (1,0) | (1,1) | (0,0) | (0,1) |

(1,1) | (1,1) | (1,0) | (0,1) | (0,0) |

Notice that this set is really like two copies of *Z*_{2} pasted next to each other. When two sets are put together to form a new set of ordered pairs in this manner, we call it a *cross product*. So this example is just the cross product of *Z*_{2} with itself, denoted *Z*_{2}* _{ }* x

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