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  2. Permutation group - Wikipedia

    en.wikipedia.org/wiki/Permutation_group

    A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]

  3. Symmetric group - Wikipedia

    en.wikipedia.org/wiki/Symmetric_group

    In the theory of Coxeter groups, the symmetric group is the Coxeter group of type A n and occurs as the Weyl group of the general linear group. In combinatorics, the symmetric groups, their elements (permutations), and their representations provide a rich source of problems involving Young tableaux, plactic monoids, and the Bruhat order.

  4. Permutation - Wikipedia

    en.wikipedia.org/wiki/Permutation

    The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition of functions (performing one rearrangement after the other), which results in another function (rearrangement).

  5. Symmetry group - Wikipedia

    en.wikipedia.org/wiki/Symmetry_group

    In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.

  6. Parity of a permutation - Wikipedia

    en.wikipedia.org/wiki/Parity_of_a_permutation

    The signature defines the alternating character of the symmetric group S n. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (ε σ), which is defined for all maps from X to X, and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as sgn(σ) = (−1) N(σ)

  7. Permutation graph - Wikipedia

    en.wikipedia.org/wiki/Permutation_graph

    Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition. [1]

  8. Inversion (discrete mathematics) - Wikipedia

    en.wikipedia.org/wiki/Inversion_(discrete...

    If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph, where an edge corresponds to the swapping of two elements on consecutive places. This Cayley graph of the symmetric group is similar to its permutohedron, but with each permutation ...

  9. Affine symmetric group - Wikipedia

    en.wikipedia.org/wiki/Affine_symmetric_group

    Each affine symmetric group is an infinite extension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as descents and inversions can be defined in the affine case. As in the finite case, the natural ...