Section 5.1
Symmetric groups
Let X and Y be sets. We recall:
Definition
A map f : X -> Y is called
A map f : X -> Y is called
- injective if f(x) = f(x') implies
x = x', for all x, x'
X;
- surjective if, for
every y Y, there
exists an element x X
with y = f(x);
- bijective if it is both injective and surjective.
We are mainly concerned with bijections of finite set X to itself. Often we work with the set X of integers from 1 to n, thus X = {1, 2, ..., n}. There is no loss of generality, since we will see soon that there is no essential difference in the naming of the elements. The advantage of the natural numbers as names of the elements of X is twofold:
- they have a natural ordering (this is convenient since we often intend to write the elements in a row);
- there is an infinite number of them (in contrast with, for example, the letters of the alphabet).
We will use no arithmetic properties of the natural numbers (as names of elements of X) apart from the ordering.