symmetric group

 

In mathematics, the symmetric group on a set X, denoted by S_X 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, S_X forms a group. The operation is also written as fg (and sometimes, although not here, as gf).

Finite symmetric groups

Of particular importance is the symmetric group on the finite set

X = {1,...,n},

denoted as S_n . The permutations 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.

Subgroups of S_n are called permutation groups.

Composing permutations

The rule of composition in the symmetric group, which is usual function composition with the symbol "o" usually omitted, is demonstrated below: Let

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),

Transpositions

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: 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.

Cycles

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 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.

Conjugacy classes

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.

Examples

Symmetric groups are Coxeter groups and reflection groups. They can be realized as a group of reflections with respect to hyperplanes . Braid groups B_n admit symmetric groups S_n as quotient groups.

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.

Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on the elements of G.

Automorphism group

For more details on this topic, see Automorphisms of the symmetric and alternating groups.

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, S_n is a complete group: its automorphism group is itself: it has no center or outer automorphisms.

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 .

See also

• Representation theory of the symmetric group
• Young tableau
• Young symmetrizer
• Dihedral group of order 6 (S₃)
• Alternating group

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