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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]
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.
Another example of a symmetry group is that of a combinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley graph; the free group is the symmetry group of an infinite tree graph.
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)
This can occur in many ways; for example, if X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object X is a set of points in the plane with its metric structure or any other metric space , a symmetry is a bijection of the set to itself which preserves the ...
If X consists of n elements and G consists of all permutations, G is the symmetric group S n; in general, any permutation group G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited any group as a permutation group, acting on itself (X = G) by means of the left regular representation.
The first group homology group (concretely, the abelianization) is (for m odd this is isomorphic to ): the factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to (concretely, by taking the product of all the values), while the sign map on the symmetric group yields the .
The alternating group, symmetric group, and their double covers are related in this way, and have orthogonal representations and covering spin/pin representations in the corresponding diagram of orthogonal and spin/pin groups. Explicitly, S n acts on the n-dimensional space R n by permuting coordinates (in matrices, as permutation matrices).