Symmetric group

The symmetric group on a set is the group of all bijections from that set to itself, with function composition as the operation. For a finite set of size n, it has n! elements and is denoted Sₙ.

Permutation: A bijection from a set to itself, often described by how it rearranges the set’s elements. Each element of a symmetric group is a permutation.
Disjoint cycle decomposition: The expression of a permutation as a product of cycles with no shared elements. It breaks each symmetric-group element into independently acting parts.
Alternating group: The subgroup of a symmetric group consisting of all even permutations. It captures the permutations of even parity within Sₙ.
Cyclic group: A group generated by a single element. Unlike Sₙ for n ≥ 3, a cyclic group has commuting elements.
Function composition: The operation of applying one function after another, producing a composite function. Composition is the group operation that combines permutations.
Transposition: A permutation that exchanges two elements and fixes all others. Transpositions generate every finite symmetric group.
Representation theory of the symmetric group: The study of linear representations of symmetric groups and their irreducible components. It translates permutation structure into matrices and connects it to partitions.
Dihedral group: The group of symmetries of a regular polygon, including rotations and reflections. It contains geometric symmetries rather than every rearrangement of the vertices.
Cycle notation: A notation that records the elements moved successively by a permutation cycle. It gives a compact way to write permutations in Sₙ.
Parity of a permutation: The classification of a permutation as even or odd according to its sign. Parity separates Sₙ into two classes and defines a subgroup.

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