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Encyclopedia > Symmetric group

In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i.e., two such functions f and g can be composed to yield a new bijective function f o g, defined by (f o g)(x) = f(g(x)) for all x in X. Using this operation, SX forms a group. The operation is also written as fg (and sometimes, but not in Wikipedia, as gf). Mathematics is the study of quantity, structure, space and change. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...


Of particular importance is the case of a finite set X = {1,...,n}, which we write as Sn. The remainder of this article will discuss Sn. The elements of Sn are called permutations; there are n! of them. The group Sn is abelian if and only if n ≤ 2. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ... In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...


Subgroups of Sn are called permutation groups. In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ...


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<j, for which f(j)<f(i). Note that for any transposition composed with f, the parity of this number changes.


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: Sn → {+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 An. It is a normal subgroup of Sn and has n! / 2 elements. The group Sn is the semidirect product of An and any subgroup generated by a single transposition. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, the permutations of a finite set (i. ... In mathematics an alternating group is the group of even permutations of a finite set. ... In 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: . There are... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...


A cycle is a permutation f for which there exists an element x in {1,...,n} such that x, f(x), f2(x), ..., fk(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 Sn can be written as a product of disjoint cycles; this representation is unique up to the order of the factors. Look up Up to in Wiktionary, the free dictionary Modern Slang In modern slang, up to means you are either willing to engage in an act (Sally is up to going to the park), capable of an act (Im sorry, Im just not up to it) or are...


The conjugacy classes of Sn correspond to the cycle structures of permutations; that is, two elements of Sn 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. In 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 groups structure. ...


Braid groups are generalizations of symmetric groups. In mathematics, the braid group on n strands, denoted by Bn, is a certain group which has a nice geometrical representation and in a sense generalizes the symmetric group Sn. ...


See also


  Results from FactBites:
 
PlanetMath: symmetric group (92 words)
We call this group the symmetric group and it is often denoted
Cross-references: cycle notation, finite, structure, group, composition, bijective functions, permutations
This is version 2 of symmetric group, born on 2003-11-10, modified 2003-11-29.
  More results at FactBites »

 
 

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