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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.
The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). [1] The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by S n, and may be called the symmetric group on n letters. By Cayley's theorem, every group is isomorphic to some ...
A similar construction with a pin group yields the 2-fold cover of the symmetric group: Pin ± (n) → O(n). As there are two pin groups, there are two distinct 2-fold covers of the symmetric group, 2⋅S ± n, also called ~ and Ŝ n.
C 1 is the trivial group containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". C 2 is the symmetry group of the letter "Z", C 3 that of a triskelion, C 4 of a swastika, and C 5, C 6, etc. are the symmetry groups of similar swastika-like figures with five, six, etc. arms instead of four.
D 2n: the dihedral group of order 2n, the same as Dih n (notation used in section List of small non-abelian groups) S n: the symmetric group of degree n, containing the n! permutations of n elements; A n: the alternating group of degree n, containing the even permutations of n elements, of order 1 for n = 0, 1, and order n!/2 otherwise; Dic n ...
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 ...
For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the order of the group (number of invariant symmetry operations). The finite group notation used is: Z n: cyclic group of order n, D n: dihedral group isomorphic to the symmetry group of an n–sided regular ...
Considering the symmetric group S n of all permutations of the set {1, ..., n}, we can conclude that the map sgn: S n → {−1, 1} that assigns to every permutation its signature is a group homomorphism. [2] Furthermore, we see that the even permutations form a subgroup of S n. [1] This is the alternating group on n letters, denoted by A n. [3]