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Home > Symmetric group

Of particular importance is the case of a finite set X = {1,...,n}, which we write as S_n. The remainder of this article will discuss S_n. The elements of S_n are called permutations; there are n! of them. The group S_n is abelian if and only if n ≤ 2.

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. 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 signature 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 subgroupIn mathematics, a normal subgroup ''N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G the element g-1ng is still in N''. The statement N is a normal subgroup of G is written: :. Another way to put t of S_n and has n! / 2 elements. The group S_n is the semidirect productIn abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. Some equivalent definitions Let G be a group, N a normal subgroup of G and H a subgroup of G''. The following statements are equi 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), f²(x), ..., f^k(x) = x are the only elements moved by f. The permutation f shown above is a cycle, since f(1) = 4, f(4) = 3 and f(3) = 1. 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 toIn mathematics, the jargon term up to xxxx" is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. xxxx" describes a property or process which transforms an element into one from the s the order of the factors.

The conjugacy classesIn mathematics, the elements of any group may be partitioned into conjugacy classes members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a group's structure. Definition Suppose G is a 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.

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Topics: Symmetric Group Permutation Product Cycle Elements Odd Written...

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