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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'').
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.
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),
:
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
:
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 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.
★ Representation theory of the symmetric group
★ Young tableau
★ Young symmetrizer
★ Dihedral group of order 6 (S3)
★ Alternating group
| Contents |
| Finite symmetric groups |
| Composing permutations |
| Transpositions |
| Cycles |
| Conjugacy classes |
| Examples |
| See also |
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
:
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''), ''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.
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.
See also
★ Representation theory of the symmetric group
★ Young tableau
★ Young symmetrizer
★ Dihedral group of order 6 (S3)
★ Alternating group
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