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Of particular importance is the symmetric group on the finite set ''X'' = {1,...,''n''}, denoted as S''n'' . The Permutation s of ''X'' form the set of bijective functions. The group S''n'' has order ''n''! and is not Abelian for ''n'' > 2. Similarly, the group S''n'' is Solvable if and only if ''n'' ≤ 4. The remainder of this article will discuss S''n''. Subgroup s of ''S''''n'' are called Permutation Group s. The rule of composition in the symmetric group is demonstrated below: Let : and : Applying ''f'' after ''g'' maps 1 to 2, and then to itself; 2 to 5 to 4; 3 to 4 to 5, and so on. So composing ''f'' and ''g'' gives : . A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation ''g'' from above can be written as ''g'' = (1 2)(2 5)(3 4). Since ''g'' can be written as a product of an odd number of transpositions, it is then called an Odd Permutation , whereas ''f'' is an ''' Even Permutation '''. The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. To see this, consider the function which maps a permutation to an integer corresponding to the number of pairs (i,j), i The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd. Thus we can define the sign of a permutation: : With this definition, :sgn: S''n'' → {+1,-1} is a Group Homomorphism ({+1,-1} is a group under multiplication, where +1 is ''e'', the Neutral Element ). The Kernel of this homomorphism, i.e. the set of all even permutations, is called the Alternating Group A''n''. It is a Normal Subgroup of S''n'' and has ''n''! / 2 elements. The group S''n'' is the Semidirect Product of A''n'' and any subgroup generated by a single transposition. A Cycle is a permutation ''f'' for which there exists an element ''x'' in {1,...,''n''} such that ''x'', ''f''(''x''), ''f''2(''x''), ..., ''f''''k''(''x'') = ''x'' are the only elements moved by ''f''. The permutation ''h'' defined by : is a cycle, since ''h''(1) = 4, ''h''(4) = 3 and ''h''(3) = 1, leaving 2 and 5 untouched. We denote such a cycle by (1 4 3). The ''length'' of this cycle is three. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are ''disjoint'' if they move different elements. Disjoint cycles commute, e.g. in S6 we have (3 1 4)(2 5 6) = (2 5 6)(3 1 4). Every element of S''n'' can be written as a product of disjoint cycles; this representation is unique Up To the order of the factors. The Conjugacy Classes of S''n'' correspond to the cycle structures of permutations; that is, two elements of S''n'' are conjugate if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S5, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. Symmetric groups are Coxeter Group s and Reflection Group s. They can be realized as a group of reflections with respect to hyperplanes . Braid Group s B''n'' contain symmetric groups S''n'' as Quotient Group s. For a list of elements of S4, see Cycle Notation . See Cube for the proper rotations of a cube, which form a group isomorphic with S4. SEE ALSO
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