PERMUTATION GROUPS

A permutation is an arrangement.

The letters a, b, c can be arranged in six ways.
e.g. abc, bca, cab, acb, cba, bac.

The more usual notation for these six permutations is:
, , , , , .

Composition of Permutations

Suppose we want to find the composition of two permutations, e.g. .

The process starts on the right, i.e. with the permutation shown in red and blue here.

Thus we get           i.e.      .

Identity and Inverses.

The identity permutation is .

To find the inverse of a permutation, e.g. , we simply rewrite it upside down, then rearrange it so that the top line is in alphabetical order.

That is, .

Larger Permutations.

The set of letters a, b, c, d can be arranged in 24 different ways.

Elements of this permuation group would be written:
, , , , , , etc.

The identity element is .

Composition and inverses behave exactly as in the previous permutation group.