Permutation

Let S be a finite set. A permutation ƒ of S is a one to one mapping from S onto S, i.e.



In other words ƒ:x (x) for ƒ(x), ∀ x S;

where (x) for ƒ(x) is called the image of x S under ƒ.

Fixed or moved element

Fixed: An element x S is said to be fixed if

(x) ƒ = x, ∀ x S.

Moved: An element x S is said to be moved by ƒ if it is not fixed by ƒ.

Identity Permutation: The identity permutation is the permutation that fixes all the elements of S. It is generally, denoted by I and

(x) I = x, ∀ x S.

In other words, a permutation which takes every element of S onto itself is called the identity permutation.

Illustration: If S = (1, 2, 3), then the identity permutation of S is

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