Pascal's Triangle and Its Relatives

Background

A Quick Sketch of Some Group Theory

With this noncommutativity in mind, in general a + H need not equal H + a.

Of course, if + is commutative then a + H = H + a for all subgroups H and all a.

Even when + is not commutative, a + H = H + a can be true for all a, for certain subgroups H of G.

These are called the normal subgroups of G.

If H is a normal subgroup of G, we can define an operation of cosets:

(a + H) + (b + H) = (a + b) + H

With this operation, the cosets of H form a group, called the quotient group G/H.

For example, consider Z6/H1.

We have seen the cosets are 0 + H1, 1 + H1, and 2 + H1. This table shows the structure of the quotient group.

+ 0 + H1 1 + H1 2 + H1 0 + H1 1 + H1 2 + H1 0 + H1 1 + H1 2 + H1 1 + H1 2 + H1 0 + H1 2 + H1 0 + H1 1 + H1

The function a + H1 -> a is an isomrphism from Z6/H1 to Z3.

Because H1 is isomorphic to Z2, we can write this isomorphism as

Z6/Z2 = Z3

Return to Some group theory.