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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.
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 ...
This finishes the proof that the Galois group of a general equation is the symmetric group, and thus proves the original Abel–Ruffini theorem, which asserts that the general polynomial equation of degree n cannot be solved in radicals for n > 4.
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 ...
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:
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 ...