4.2 Cycle Notation and Conjugacy Classes

Cycle notation simplifies how we represent permutations in symmetric groups. It's a handy tool that lets us see how elements move around. By using cycles, we can easily spot patterns and understand the structure of permutations.

Conjugacy classes group permutations with similar cycle structures. This concept is crucial for understanding symmetric groups and their properties. It helps us organize permutations and reveals important relationships between them, making complex group theory more manageable.

Cycle Notation for Permutations

Representing Permutations

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• Express permutations as bijective functions from a set to itself using cycle notation
• Write permutations as a product of disjoint cycles in cycle notation
  • Each cycle represents the movement of elements within the set
  • The order of cycles in the product does not matter, but the order of elements within each cycle is important
• Define the length of a cycle as the number of elements it contains
  • A cycle of length 1 is called a fixed point (e.g., (1))

Properties of Cycles

• Identify disjoint cycles as cycles with no elements in common
• Understand that the order of the cycles in the product does not affect the permutation
• Recognize that the order of elements within each cycle is crucial for the permutation
• Determine the number of elements moved by a permutation based on its cycle structure
  • Elements not appearing in any cycle are fixed points

Parity of Permutations

Determining Parity

• Define the parity of a permutation as either even or odd
  • Parity depends on the number of transpositions (cycles of length 2) needed to express the permutation
• Express permutations as a product of transpositions
  • An even permutation can be expressed as a product of an even number of transpositions
  • An odd permutation can be expressed as a product of an odd number of transpositions
• Determine the parity of a permutation based on its cycle structure
  • A permutation is even if and only if it has an even number of cycles of even length

Properties of Parity

• Define the sign of a permutation as +1 for even permutations and -1 for odd permutations
• Understand that the composition of two even or two odd permutations is even, while the composition of an even and an odd permutation is odd
• Apply parity to solve problems involving permutations and their properties
  • For example, determine the sign of a permutation given its cycle structure

Conjugacy Classes in Symmetric Groups

Defining Conjugacy

• Define conjugate permutations α and β in the symmetric group Sn
  • There exists a permutation σ in Sn such that σ⁻¹ασ = β
• Define the conjugacy class of a permutation α as the set of all permutations conjugate to α, denoted by [α]
• Understand that conjugacy is an equivalence relation on the symmetric group
  • Conjugacy classes form a partition of the symmetric group

Properties of Conjugacy Classes

• Recognize that permutations in the same conjugacy class have the same cycle structure
  • They have the same number of cycles of each length
• Calculate the number of permutations in a conjugacy class using the index of the centralizer of any permutation in the class
• Apply conjugacy classes to solve problems involving permutations and their properties
  • For example, determine the number of conjugacy classes in Sn for a given n

Conjugacy Class Equation for Symmetric Groups

Proving the Equation

• State the conjugacy class equation
  • The sum of the indices of the centralizers of representatives from each conjugacy class in the symmetric group Sn is equal to n!, the order of Sn
• Prove the conjugacy class equation using the following steps:
  1. Show that the centralizer of a permutation α in Sn is isomorphic to a direct product of symmetric groups, one for each cycle in α
  2. Determine the order of the centralizer of α using the orders of the symmetric groups in the direct product
  3. Calculate the index of the centralizer of α by dividing the order of Sn by the order of the centralizer
  4. Sum the indices of the centralizers of representatives from each conjugacy class and show that the result is equal to n!

Applications of the Equation

• Use the conjugacy class equation to count the number of conjugacy classes in the symmetric group Sn
• Apply the conjugacy class equation to determine the number of permutations in each conjugacy class
• Solve problems involving the structure and properties of the symmetric group using the conjugacy class equation
  • For example, find the number of self-conjugate permutations in Sn