4.1 Structure and Properties of the Symmetric Group

The symmetric group is a fundamental concept in algebra, representing all possible ways to rearrange elements in a set. It's crucial for understanding permutations and their properties, forming the basis for many advanced topics in group theory and combinatorics.

Exploring the structure and properties of the symmetric group reveals its rich mathematical characteristics. From closure and associativity to identity and inverse elements, these properties showcase the group's versatility and importance in various mathematical applications.

The Symmetric Group

Definition and Notation

Top images from around the web for Definition and Notation

• 

  Permutation group - Knowino View original

  Is this image relevant?

• 

  Symmetric group S4 - Wikiversity View original

  Is this image relevant?

• 

  Combinatorics Overview View original

  Is this image relevant?

• 

  Permutation group - Knowino View original

  Is this image relevant?

• 

  Symmetric group S4 - Wikiversity View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Notation

• 

  Permutation group - Knowino View original

  Is this image relevant?

• 

  Symmetric group S4 - Wikiversity View original

  Is this image relevant?

• 

  Combinatorics Overview View original

  Is this image relevant?

• 

  Permutation group - Knowino View original

  Is this image relevant?

• 

  Symmetric group S4 - Wikiversity View original

  Is this image relevant?

1 of 3

• The symmetric group on a set X, denoted as Sym(X) or S(X), is the group of all bijections from X to itself under the operation of function composition
• When X is a finite set with n elements, the symmetric group is denoted as Sn and consists of all permutations of the set {1, 2, ..., n}
• Examples of symmetric groups:
  • S3 is the symmetric group on the set {1, 2, 3} and has 6 elements
  • S4 is the symmetric group on the set {1, 2, 3, 4} and has 24 elements

Elements and Representations

• Elements of the symmetric group are called permutations, which can be represented using cycle notation or as products of transpositions
  • Cycle notation: (1 2 3) represents the permutation that maps 1 to 2, 2 to 3, and 3 to 1
  • Product of transpositions: (1 2)(2 3) represents the permutation that first swaps 1 and 2, then swaps 2 and 3
• The degree of a permutation is the number of elements in the set X
• The order of a permutation is the least positive integer k such that the permutation composed with itself k times equals the identity permutation
  • Example: The permutation (1 2 3) has order 3 because (1 2 3)^3 = (1)(2)(3), the identity permutation

Properties of the Symmetric Group

Closure and Associativity

• Closure: The composition of any two permutations in Sn results in another permutation in Sn, proving that the symmetric group is closed under function composition
  • Example: In S3, (1 2) ∘ (2 3) = (1 2 3), which is also a permutation in S3
• Associativity: The composition of permutations is associative, meaning that (α ∘ β) ∘ γ = α ∘ (β ∘ γ) for any permutations α, β, and γ in Sn
  • Example: In S4, ((1 2) ∘ (3 4)) ∘ (2 3) = (1 2) ∘ ((3 4) ∘ (2 3))

Identity and Inverse

• Identity: The identity element in Sn is the identity permutation, denoted as ε or (1), which maps each element to itself. Composing any permutation with the identity results in the original permutation
  • Example: In S3, (1 2 3) ∘ (1) = (1 2 3) and (1) ∘ (1 2 3) = (1 2 3)
• Inverse: For every permutation α in Sn, there exists a unique inverse permutation α^(-1) such that α ∘ α^(-1) = α^(-1) ∘ α = ε
  • The inverse of a permutation can be found by reversing the order of the elements in each cycle or by writing the permutation as a product of transpositions and reversing their order
  • Example: In S3, the inverse of (1 2 3) is (3 2 1), and (1 2 3) ∘ (3 2 1) = (1)(2)(3)

Order of the Symmetric Group

Order of Sn

• The order of the symmetric group Sn, denoted as |Sn|, is equal to n!, the number of distinct permutations of n elements
  • Example: |S3| = 3! = 6 and |S4| = 4! = 24

Subgroups and Lagrange's Theorem

• Subgroups of the symmetric group include cyclic groups generated by a single permutation, dihedral groups, and alternating groups
• The order of a subgroup divides the order of the symmetric group, as stated by Lagrange's theorem
  • Example: The cyclic subgroup generated by (1 2 3) in S3 has order 3, which divides the order of S3 (6)
• The order of a cyclic subgroup generated by a permutation α is equal to the order of α, which is the least common multiple of the lengths of the disjoint cycles in its cycle decomposition
  • Example: The permutation (1 2)(3 4 5) has order lcm(2, 3) = 6

Generators of the Symmetric Group

Generating Sets

• A generating set for a group is a subset of elements that can generate the entire group through the group operation
• The symmetric group Sn can be generated by a set of transpositions, specifically the adjacent transpositions (1 2), (2 3), ..., (n-1 n)
  • Example: S3 can be generated by the adjacent transpositions (1 2) and (2 3)
• Another generating set for Sn consists of a transposition and an n-cycle, such as (1 2) and (1 2 ... n)
  • Example: S4 can be generated by the transposition (1 2) and the 4-cycle (1 2 3 4)

Minimal Number of Generators

• The minimal number of generators required for Sn is 2
  • One generator can be a transposition, and the other an n-cycle
  • Example: S5 can be minimally generated by the transposition (1 2) and the 5-cycle (1 2 3 4 5)