Pascal's Triangle and Its Relatives

Background

A Quick Sketch of Some Group Theory

We have seen that the subgroup H1 = {0, 3} of Z6 is isomprphic to Z2.

Note that 0 + H1 = {0 + 0, 0 + 3} = {0, 3} 1 + H1 = {1 + 0, 1 + 3} = {1, 4} 2 + H1 = {2 + 0, 2 + 3} = {2, 5} form a partition of Z6.

That is, the union of these sets is all of Z6 and each set has no element in common with any other set.

We say 0 + H1, 1 + H1, and 2 + H1,are the cosets of H1.

What about 3 + H1, 4 + H1, and 5 + H1?

It is not difficult to see that 3 + H1 = 0 + H1, 4 + H1 = 1 + H1,and 5 + H1 = 2 + H1.

In fact, this generalizes. For any subgroup H of any group G, if b is in the coset a + H, then b + H = a + H.

For this reason, an element of a coset is called a representative of that coset.

Return to Some group theory.