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The symmetric group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois theory. In invariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric functions.
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]
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).
These identities may be derived by enumerating permutations directly. For example, a permutation of n elements with n − 3 cycles must have one of the following forms: n − 6 fixed points and three two-cycles; n − 5 fixed points, a three-cycle and a two-cycle, or; n − 4 fixed points and a four-cycle.
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 ...
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 ...
For every symmetric group other than S 6, there is no other conjugacy class consisting of elements of order 2 that has the same number of elements as the class of transpositions. Or as follows: Each permutation of order two (called an involution) is a product of k > 0 disjoint transpositions, so that it has cyclic structure 2 k 1 n−2k.
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.