An even permutation is a permutation of a finite set that can be expressed as the product of an even number of transpositions, which are swaps of two elements. In the context of group theory, the set of all even permutations of a finite set forms a subgroup known as the alternating group, which is significant in the study of symmetric groups and their properties.

Even Permutation

A finite set is a set with a limited number of elements, allowing for complete enumeration and analysis of its members. This concept is fundamental in mathematics, particularly in set theory, as it contrasts with infinite sets and facilitates the application of counting principles and combinatorial techniques.

Finite Set

Product of an even number of transpositions

Swaps of two elements

Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.

Group Theory

Set of all even permutations

A subgroup is a subset of a group that itself forms a group under the same operation as the original group. It must satisfy the group axioms: closure, associativity, identity, and invertibility within the subset.

Subgroup

The alternating group, denoted as A_n, is the group of even permutations of a finite set of n elements, forming a normal subgroup of the symmetric group S_n with half its order. It is a fundamental object in group theory, playing a crucial role in the classification of simple groups and providing insights into the symmetry properties of mathematical structures.

Alternating Group

Symmetric groups, denoted as S_n, are fundamental algebraic structures in group theory that consist of all permutations of a finite set of n elements, capturing the essence of symmetry in mathematical objects. They play a critical role in various fields such as algebra, combinatorics, and geometry, serving as a cornerstone for understanding more complex groups and their representations.

Concepts

Relevant Degrees

Your Lessons