Some examples of groups

The symmetric group --------------------

A permutation (also called a substitution) of a set E with n elements is a bijection from the set to itself. A permutation simply rearranges the elements. To study permutations, we generally use the set of integers {1, 2, 3, …, n}. Any set E under consideration can be reduced to this case by numbering its elements.

All the permutations of {1, 2, 3, …, n} form a group under composition, called the symmetric group of degree n; it is denoted by σ*_n or S_n *.

This group is finite:

it contains n! = n × (n − 1) × (n − 2) ×… × 2 × 1 elements. There are n possible choices for the image of 1, leaving n − 1 choices for the image of 2, and so on; by the time we reach n, only one possible image remains. This number, called n factorial, grows very rapidly with n: σ₃ has six elements, σ₄ has 24 and σ₁₀ has 3,628,800.