Introduction to Permutations
A permutation of a set is a rearrangement of its elements. For a finite set \( S \) with \( n \) elements, a permutation can be thought of as a bijective function \( \sigma: S \rightarrow S \) that reorders the elements of \( S \). The total number of permutations of a set of size \( n \) is given by \( n! \) (n factorial).
Types of Permutations:
1. Even Permutations: A permutation that can be expressed as an even number of transpositions (two-element swaps).
2. Odd Permutations: A permutation that can be expressed as an odd number of transpositions.
The set of all permutations of a set \( S \) of size \( n \) forms a group under the operation of composition, known as the symmetric group, denoted as \( S_n \).
Definition of Permutation Groups
A permutation group is a group whose elements are permutations of a given set and whose operation is the composition of these permutations. Formally, a permutation group \( G \) on a set \( S \) can be defined as a subset of the symmetric group \( S_n \), where \( n = |S| \).
Key Properties of Permutation Groups:
- Closure: The composition of any two permutations in the group is also a permutation in the group.
- Identity Element: The identity permutation, which leaves every element in its original position, is an element of the group.
- Inverses: For every permutation in the group, an inverse permutation exists that undoes the effect of the original permutation.
Examples of Permutation Groups
Permutation groups can take various forms, depending on the set being permuted and the specific permutations included in the group.
The Symmetric Group \( S_n \)
The symmetric group \( S_n \) is the group of all permutations of a set with \( n \) elements. It has the following properties:
- Order: The order of \( S_n \) is \( n! \).
- Structure: \( S_n \) is non-Abelian for \( n \geq 3 \), meaning that the composition of two permutations may not be commutative.
Example: The symmetric group \( S_3 \) consists of the following permutations of the set \( \{1, 2, 3\} \):
- \( e \): the identity permutation (1, 2, 3)
- \( (1\ 2) \): swaps 1 and 2
- \( (1\ 3) \): swaps 1 and 3
- \( (2\ 3) \): swaps 2 and 3
- \( (1\ 2\ 3) \): cycles 1 to 2, 2 to 3, and 3 to 1
- \( (1\ 3\ 2) \): cycles 1 to 3, 3 to 2, and 2 to 1
The Alternating Group \( A_n \)
The alternating group \( A_n \) is the group of all even permutations of a set with \( n \) elements. It is a normal subgroup of the symmetric group \( S_n \) and has the following properties:
- Order: The order of \( A_n \) is \( \frac{n!}{2} \).
- Structure: \( A_n \) is non-Abelian for \( n \geq 3 \).
Example: For \( n = 3 \), the group \( A_3 \) consists of the following permutations:
- \( e \): the identity (1, 2, 3)
- \( (1\ 2\ 3) \): an even permutation
- \( (1\ 3\ 2) \): another even permutation
Subgroups of Permutation Groups
Permutation groups can contain several interesting subgroups, which can help in understanding their structure and properties.
Transitive Groups
A permutation group is said to be transitive if there is only one orbit when the group acts on the set, meaning that it can move any element of the set to any other element.
Example: The symmetric group \( S_n \) is transitive because any element can be mapped to any other element through some permutation.
Point Stabilizers
For a given element \( x \) in the set \( S \), the stabilizer of \( x \) in a permutation group \( G \) is the subgroup of \( G \) that keeps \( x \) fixed. This subgroup is denoted \( G_x \) and consists of all permutations \( \sigma \) in \( G \) such that \( \sigma(x) = x \).
Properties of Stabilizers:
- The size of the orbit of \( x \) under the action of \( G \) can be computed using the Orbit-Stabilizer Theorem, which states:
\[
|G| = |G_x| \cdot |G \cdot x|
\]
Applications of Permutation Groups
Permutation groups have far-reaching applications across various disciplines:
Combinatorics
Permutation groups are used extensively in combinatorial problems, such as counting the number of distinct arrangements of objects and solving problems related to symmetry.
Cryptography
Certain cryptographic systems utilize permutation groups in their algorithms to secure data. The complexity of permutations makes them suitable for cryptographic functions.
Physics and Chemistry
Permutation groups play a role in understanding symmetries in physical systems, such as particle arrangements in quantum mechanics and the geometric structures of molecules in chemistry.
Conclusion
Permutation groups are a central topic in abstract algebra, providing deep insights into the nature of symmetry and structure in mathematical systems. Their rich theory extends beyond mere abstract definitions, with practical implications in multiple fields of study. Understanding permutation groups—whether through the lens of symmetric groups, alternating groups, or their various substructures—opens up avenues for exploration in both pure and applied mathematics. As we continue to delve into the complexities of group theory, the significance of permutation groups remains ever-present, influencing not only mathematical thought but also the practical applications that shape our understanding of the world.
Frequently Asked Questions
What is a permutation group in abstract algebra?
A permutation group is a mathematical structure that consists of a set of permutations of a given set, along with the operation of composition of these permutations. It forms a group under this operation, meaning it satisfies the group properties: closure, associativity, identity, and invertibility.
How do you determine if a set of permutations forms a group?
To determine if a set of permutations forms a group, check if the set is closed under composition, contains the identity permutation, and that every permutation has an inverse in the set. Additionally, composition must be associative.
What is the symmetric group and its significance?
The symmetric group, denoted as Sn, is the group of all possible permutations of n elements. It is significant because it serves as a fundamental example of a permutation group and is used in various fields such as combinatorics, group theory, and algebra.
What are even and odd permutations?
Even permutations are those that can be expressed as a product of an even number of transpositions, while odd permutations can be expressed as a product of an odd number of transpositions. This classification is important for understanding the structure of the symmetric group.
What is the cycle notation for permutations?
Cycle notation is a way to represent permutations by showing the orbits of elements under the permutation. For example, the permutation that sends 1 to 2, 2 to 3, and 3 back to 1 can be written as (1 2 3), indicating a cycle of length 3.
How are permutation groups used in combinatorial problems?
Permutation groups are used in combinatorial problems to analyze the symmetries of objects, count distinct arrangements, and solve problems related to counting and organizing structures. They help in understanding the underlying symmetry and organizing principles in combinatorial designs.
