Pascal's Triangle and Its Relatives

Background

A Quick Sketch of Some Group Theory

If G consists of a finite collection of elements, it is called a finite group and the number of elements is the order of G, denoted |G|.

The order of an element g in G is the smallest number n for which g + ... + g = e, where n is the number of copies of g in the sum.

Consider Z6, the integers mod 6, as an illustration.

Closure and associativity are easy to verify.

The identity element is 0: for example, 0 + 1 = 1 + 0 = 1.

The inverse of each element a is 6 - a: for example, 1 + 5 is congruent to 0 (mod 6).

So Z6 is a group under the operation of addition. Its order is 6.

In Z6 the element 2 has order three: 2 + 2 + 2 is congruent to 0 (mod 6).

Return to Some group theory.