Permutation and Symmetric Groups
Introduction to Permutations
Permutation is an arrangement of objects in a specific order. In mathematics, permutation refers to the act of rearranging elements of a set into a particular sequence or order. Let's dive into the details of permutations and their applications.
Definition of Permutation
A permutation of a set is an ordered arrangement of its elements. For example, consider a set {1, 2, 3}. The permutations of this set are:
- {1, 2, 3}
- {1, 3, 2}
- {2, 1, 3}
- {2, 3, 1}
- {3, 1, 2}
- {3, 2, 1}
Permutation Notation
Permutations are often denoted using cycle notation or as a product of disjoint cycles. For instance, the permutation (123) represents the cyclic permutation of {1, 2, 3} as {1, 2, 3} → {2, 3, 1}. Similarly, (12)(34) represents the permutation of {1, 2, 3, 4} where 1 is mapped to 2, 2 is mapped to 1, 3 is mapped to 4, and 4 is mapped to 3.
Symmetric Groups
Symmetric groups are groups formed by permutations of a set. Let's explore symmetric groups and their properties.
Definition of Symmetric Group
The symmetric group, denoted by (S_n), is the group of all permutations of a set with (n) elements. It consists of all possible permutations of the elements of the set.
Properties of Symmetric Groups
1. Order of Symmetric Group: The order of the symmetric group (S_n) is (n!), where (n) is the number of elements in the set.
2. Closure Property: Symmetric groups are closed under composition, i.e., the composition of two permutations results in another permutation in the group.
3. Identity Element: The identity element of the symmetric group is the permutation that leaves all elements unchanged.
4. Inverse Element: Each permutation in the symmetric group has an inverse, which when composed with the permutation results in the identity permutation.
Example of Symmetric Group
Consider the set {1, 2, 3}. The symmetric group (S_3) consists of all possible permutations of this set:
- (123)
- (132)
- (12)
- (13)
- (23)
- (1)
- (2)
- (3)
These permutations form the symmetric group (S_3) with 6 elements.