Alternating groups are the group of even permutations of a finite set and are important in the study of symmetric groups and group theory. They are simple for n ≥ 5, meaning they have no nontrivial normal subgroups, which makes them fundamental in the classification of finite simple groups.

Alternating Groups

Even permutations

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

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.

Symmetric Groups

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

N ≥ 5

Nontrivial normal subgroups

The Classification of Finite Simple Groups is a monumental achievement in mathematics, providing a comprehensive list of all possible finite simple groups, which are the building blocks of all finite groups. This classification includes 18 infinite families and 26 sporadic groups, and its proof spans thousands of pages contributed by numerous mathematicians over several decades.

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