Permutation group

Definition

A permutation group is a mathematical structure consisting of a set of permutations that can be combined through composition and that satisfies the group properties: closure, associativity, identity, and invertibility. This concept is essential for understanding the symmetries and transformations of objects, as it connects to the study of groups by illustrating how elements can be rearranged in systematic ways.

5 Must Know Facts For Your Next Test

1. A permutation group can be finite or infinite, depending on the number of elements being permuted.
2. The order of a permutation group is defined as the number of permutations it contains, which gives insights into its structure and complexity.
3. Every permutation can be expressed as a product of disjoint cycles, which simplifies the understanding of how permutations act on elements.
4. The Cayley's theorem states that every group can be represented as a subgroup of a symmetric group, indicating that permutation groups are fundamental in group theory.
5. In permutation groups, the inverse of a permutation undoes its effect, ensuring that every element has an associated inverse within the group.

Review Questions

• How does a permutation group illustrate the properties of a mathematical group?
  • A permutation group exemplifies the properties of a mathematical group by demonstrating closure through the composition of permutations, associativity in how permutations can be grouped during composition, the existence of an identity permutation that leaves all elements unchanged, and invertibility where each permutation has an inverse that reverses its action. These properties are foundational in establishing the structure and behavior of groups in abstract algebra.
• In what ways does the concept of a symmetric group enhance our understanding of permutation groups?
  • The concept of a symmetric group enhances our understanding of permutation groups by serving as a prime example where every possible arrangement or reordering of a finite set is represented. This highlights not only the vastness and richness of permutation groups but also establishes a baseline for studying their subgroups and identifying different types of permutations such as even and odd permutations. Furthermore, it allows for exploring advanced topics such as cosets and normal subgroups within these structures.
• Evaluate how the Orbit-Stabilizer Theorem provides insight into the structure and behavior of permutation groups.
  • The Orbit-Stabilizer Theorem provides profound insight into the structure and behavior of permutation groups by linking the sizes of orbits and stabilizers, thereby facilitating an understanding of how elements interact within these groups. By determining the size of an orbit corresponding to an element under a group action, one can deduce information about its stabilizer subgroup, which retains certain properties unchanged. This relationship simplifies complex problems in counting and analyzing symmetries, showcasing how permutation groups organize mathematical objects in structured ways.

Related terms

Symmetric Group: The symmetric group is the group of all possible permutations of a finite set, with the group operation being the composition of these permutations.

Transposition: A transposition is a specific type of permutation that swaps two elements while leaving all others unchanged.

Orbit-Stabilizer Theorem: This theorem relates the size of an orbit of an element under a group action to the size of its stabilizer subgroup, providing insight into the structure of permutation groups.