Permutation group

A permutation group is a group whose elements are bijections of a set to itself, with composition as the group operation.

Permutation: A bijection from a set to itself, often represented by rearranging its elements. Each element of a permutation group is a permutation of the underlying set.
Orbit (group theory): The set of points obtainable from one point by applying elements of a group action. Orbits partition the set into regions that the permutation group can reach internally.
Dihedral group: The symmetry group of a regular polygon, generated by rotations and reflections. Its elements act as permutations of the polygon’s vertices.
Abstract group: A group considered through its operation and axioms, without a particular realization as transformations. A permutation group is a concrete realization, while the abstract group records structure up to isomorphism.
Group (mathematics): A set with an associative operation, an identity element, and an inverse for every element. Permutation groups satisfy the group axioms under composition.
Stabilizer (group theory): The subgroup consisting of elements that leave a specified point fixed under a group action. A point’s stabilizer measures which permutations preserve that point.
Alternating group: The group of even permutations on a finite set. It is a fundamental subgroup of the symmetric group.
Transformation group: A group whose elements act as transformations on a space or set. Permutation groups are transformation groups whose transformations are bijections of a set.
Function composition: An operation that applies one function and then another to the result. Composing permutations supplies the group operation.
Cycle notation: A notation that records the cyclic movement of elements under a permutation. Cycles make individual permutations and their composition easier to inspect.

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