symmetric group
n.
A group consisting of all possible permutations of a given number of items.
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A group consisting of all possible permutations of a given number of items.
In mathematics, the symmetric group on a set
X, denoted by SX or Sym(X), is the
Of particular importance is the symmetric group on the finite set
denoted as Sn . The permutations of X form the set of bijective
functions. The group Sn has order
The rule of composition in the symmetric group, which is usual

and

Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing f and g gives
.It is an easy exercise to show that a cycle of length L =k·m, taken to the k-th power, will decompose into k cycles of length m: For example (k=2, m=3),

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,
is a group homomorphism ({+1,-1} is a group under multiplication, where +1 is
e, the
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 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
Sn can be written as a product of disjoint cycles; this representation is unique
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.
Symmetric groups are Coxeter groups and
. Braid groups Bn admit symmetric groups Sn as
For a list of elements of S4, see
| n | Aut(Sn) | Out(Sn) |
| n≠2,6 | Sn | 1 |
| n = 2 | 1 | 1 |
| n = 6 | Failed to parse (unknown function\rtimes): S_6 \rtimes C_2 | C2 |
For n≠2,6, Sn is a
For n = 2, the automorphism group is trivial, but S2 is not trivial (it is isomorphic to C2, which is abelian).
For n = 6, it has an outer automorphism of order 2: Out(S6) = C2, and the automorphism group is a semidirect product: Failed to parse (unknown function\rtimes): \mbox{Aut}(S_6)=S_6 \rtimes C_2 .
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