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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 = 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.
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 symmetry group of a snowflake is D 6, a dihedral symmetry, the same as for a regular hexagon. In mathematics, a dihedral group is the group of symmetries of a regular polygon, [1][2] which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory and ...
Automorphisms of the symmetric and alternating groups. In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S 6, the ...
A 4 is isomorphic to PSL 2 (3) [1] and the symmetry group of chiral tetrahedral symmetry. A 5 is isomorphic to PSL 2 (4), PSL 2 (5), and the symmetry group of chiral icosahedral symmetry. (See [1] for an indirect isomorphism of PSL 2 (F 5) → A 5 using a classification of simple groups of order 60, and here for a direct proof). A 6 is ...
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. [1] More specifically, G is isomorphic to a subgroup of the symmetric group whose elements are the permutations of the underlying set of G. Explicitly, The homomorphism can also be understood as ...