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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.
Every symmetric group has a one-dimensional representation called the trivial representation, where every element acts as the one by one identity matrix. For n ≥ 2 , there is another irreducible representation of degree 1, called the sign representation or alternating character , which takes a permutation to the one by one matrix with entry ...
Automorphisms of the symmetric and alternating groups; Block (permutation group theory) Cayley's theorem; Cycle index; Frobenius group; Galois group of a polynomial; Jucys–Murphy element; Landau's function; Oligomorphic group; O'Nan–Scott theorem; Parker vector; Permutation group; Place-permutation action; Primitive permutation group; Rank ...
Pages in category "Permutation groups" ... Affine symmetric group; Alternating group; Automorphisms of the symmetric and alternating groups; B. Base (group theory) ...
The Hamiltonian cycle in the Cayley graph of the symmetric group generated by the Steinhaus–Johnson–Trotter algorithm Wheel diagram of all permutations of length = generated by the Steinhaus-Johnson-Trotter algorithm, where each permutation is color-coded (1=blue, 2=green, 3=yellow, 4=red).
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)
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 .
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