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The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that of function composition. [1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree is the symmetric group on the set .
The group of isometries of space induces a group action on objects in it, and the symmetry group Sym (X) consists of those isometries which map X to itself (as well as mapping any further pattern to itself). We say X is invariant under such a mapping, and the mapping is a symmetry of X. The above is sometimes called the full symmetry group of X ...
A cyclic permutation consisting of a single 8-cycle. There is not widespread consensus about the precise definition of a cyclic permutation. Some authors define a permutation σ of a set X to be cyclic if "successive application would take each object of the permuted set successively through the positions of all the other objects", [1] or, equivalently, if its representation in cycle notation ...
the product of two 2-cycles such as (1 2)(3 4) maps to another product of two 2-cycles such as (3 5)(4 6), accounting for 45 permutations; the product of a 2-cycle and a 4-cycle such as (1 2 3 4)(5 6) maps to another such permutation such as (1 4 2 6)(3 5), accounting for the 90 remaining permutations. And the odd part is also conserved:
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.
The symmetric group of degree n ≥ 4 has Schur covers of order 2⋅n! There are two isomorphism classes if n ≠ 6 and one isomorphism class if n = 6. The alternating group of degree n has one isomorphism class of Schur cover, which has order n! except when n is 6 or 7, in which case the Schur cover has order 3⋅n!.
The affine symmetric group is a subgroup of the extended affine symmetric group. The extended group is isomorphic to the wreath product . Its elements are extended affine permutations: bijections such that for all integers x. Unlike the affine symmetric group, the extended affine symmetric group is not a Coxeter group.
The Weyl group of SO(2n + 1) is the semidirect product {} of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each {±1} factor of {±1} n acts on the corresponding circle factor of T × {1} by inversion, and the symmetric group S n acts on both {±1} n and T × {1} by permuting factors.