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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 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!.
For n > 4, the symmetric group of degree n has only the alternating group as a nontrivial normal subgroup (see Symmetric group § Normal subgroups). For n > 4 , the alternating group A n {\displaystyle {\mathcal {A}}_{n}} is simple (that is, it does not have any nontrivial normal subgroup) and not abelian .
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 ...
Finally the product p 1 e k−1 for i = 1 gives contributions to r(i + 1) = r(2) like for other values i < k, but the remaining contributions produce k times each monomial of e k, since any one of the variables may come from the factor p 1; thus
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:
For n = 3, 4 there are two additional one-dimensional irreducible representations, corresponding to maps to the cyclic group of order 3: A 3 ≅ C 3 and A 4 → A 4 /V ≅ C 3. For n ≥ 7 , there is just one irreducible representation of degree n − 1 , and this is the smallest degree of a non-trivial irreducible representation.
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.