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 + H11 + H12 + H1
0 + H1
1 + H1
2 + H1
0 + H11 + H12 + H1
1 + H12 + H10 + H1
2 + H10 + H11 + 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.