Permutation group

Definition

A permutation group is a mathematical structure consisting of a set of permutations (or rearrangements) of a finite set, along with the operation of composition of these permutations. This group is crucial because it captures the concept of symmetry and can be studied using the fundamental properties of groups, such as closure, identity, and inverses, which are essential for understanding the broader implications in group theory.

5 Must Know Facts For Your Next Test

1. The order of a permutation group is the total number of distinct permutations that can be formed from the elements of the set.
2. Permutation groups can be classified into various types based on their structure, such as transitive groups, which act on a single orbit.
3. The composition of two permutations within a permutation group always results in another permutation within the same group, satisfying the closure property.
4. The identity permutation, which leaves all elements unchanged, serves as the identity element in a permutation group.
5. Every permutation has an inverse permutation that reverses its effect, ensuring that every element in a permutation group has an inverse.

Review Questions

• How do permutation groups illustrate the fundamental properties of groups?
  • Permutation groups exemplify the fundamental properties of groups through their operations. Each permutation can be composed with another to yield another permutation, demonstrating closure. The presence of an identity permutation ensures that there is an element that leaves all others unchanged. Additionally, for every permutation in the group, there exists an inverse that undoes its effect, fulfilling the group's requirement for inverses.
• In what ways can cycle notation enhance our understanding of permutation groups?
  • Cycle notation enhances understanding by simplifying the representation of permutations within a group. Instead of writing out each individual mapping, cycle notation compactly expresses how elements are permuted in cycles. This makes it easier to visualize and analyze the structure of the permutation group, especially when determining properties like order or identifying conjugate elements.
• Evaluate how the concept of transitive groups contributes to our understanding of symmetry within permutation groups.
  • Transitive groups deepen our understanding of symmetry by illustrating how a permutation group can act uniformly across a set. A transitive permutation group means there is only one orbit under its action, indicating that any element can be transformed into any other through some permutation. This highlights intrinsic symmetries and structures within sets, allowing for a richer exploration of symmetrical properties across various fields such as geometry and algebra.

Related terms

Symmetric group: The symmetric group is the group of all permutations of a finite set, typically denoted as $$S_n$$ for a set of size $$n$$.

Cycle notation: Cycle notation is a way of representing permutations by indicating the cycles in which elements are permuted, which provides an efficient way to express complex permutations.

Group action: A group action describes how a group operates on a set, allowing for the study of symmetries and transformations within that set.