![Symmetric group 4; permutation list](https://upload.wikimedia.org/wikipedia/commons/3/3d/Symmetric_group_4%3B_permutation_list.svg)
In
mathematics, when ''X'' is a
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. T ...
with at least two elements, the
permutations of ''X'' (i.e. the
bijective function
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
s from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any
total ordering
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
of ''X'' is fixed, the parity (oddness or evenness) of a permutation
of ''X'' can be defined as the parity of the number of
inversions for ''σ'', i.e., of pairs of elements ''x'', ''y'' of ''X'' such that and .
The sign, signature, or signum of a permutation ''σ'' is denoted sgn(''σ'') and defined as +1 if ''σ'' is even and −1 if ''σ'' is odd. The signature defines the alternating
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
S
''n''. Another notation for the sign of a permutation is given by the more general
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
(''ε''
''σ''), which is defined for all maps from ''X'' to ''X'', and has value zero for
non-bijective maps.
The sign of a permutation can be explicitly expressed as
:
where ''N''(''σ'') is the number of
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
s in ''σ''.
Alternatively, the sign of a permutation ''σ'' can be defined from its decomposition into the product of
transpositions as
:
where ''m'' is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is
well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
.
[Jacobson (2009), p. 50.]
Example
Consider the permutation ''σ'' of the set defined by
and
In
one-line notation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
, this permutation is denoted 34521. It can be obtained from the
identity permutation
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...
12345 by three transpositions: first exchange the numbers 2 and 4, then exchange 3 and 5, and finally exchange 1 and 3. This shows that the given permutation ''σ'' is odd. Following the method of the
cycle notation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
article, this could be written, composing from left to right, as
:
There are many other ways of writing ''σ'' as a
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of transpositions, for instance
:,
but it is impossible to write it as a product of an even number of transpositions.
Properties
The identity permutation is an even permutation.
An even permutation can be obtained as the composition of an
even number and only an even number of exchanges (called
transpositions) of two elements, while an odd permutation can be obtained by (only) an odd number of transpositions.
The following rules follow directly from the corresponding rules about addition of integers:
* the composition of two even permutations is even
* the composition of two odd permutations is even
* the composition of an odd and an even permutation is odd
From these it follows that
* the inverse of every even permutation is even
* the inverse of every odd permutation is odd
Considering the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
S
''n'' of all permutations of the set , we can conclude that the map
:
that assigns to every permutation its signature is a
group homomorphism
In mathematics, 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) \cdot h(v)
w ...
.
Furthermore, we see that the even permutations form a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of S
''n''.
This is the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
on ''n'' letters, denoted by A
''n''.
[Jacobson (2009), p. 51.] It is the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
of the homomorphism sgn. The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
of A
''n'' (in S
''n'').
[Meijer & Bauer (2004), p. 72/ref>
If , then there are just as many even permutations in S''n'' as there are odd ones;] consequently, A''n'' contains ''n''!/2 permutations. (The reason is that if ''σ'' is even then is odd, and if ''σ'' is odd then is even, and these two maps are inverse to each other.)
A cycle is even if and only if its length is odd. This follows from formulas like
:
In practice, in order to determine whether a given permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation is odd if and only if this factorization contains an odd number of even-length cycles.
Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value of the determinant is the same as the parity of the permutation.
Every permutation of odd order must be even. The permutation in A4 shows that the converse is not true in general.
Equivalence of the two definitions
This section presents proofs that the parity of a permutation ''σ'' can be defined in two equivalent ways:
* as the parity of the number of inversions in ''σ'' (under any ordering); or
* as the parity of the number of transpositions that ''σ'' can be decomposed to (however we choose to decompose it).
Other definitions and proofs
The parity of a permutation of points is also encoded in its cycle structure.
Let ''σ'' = (''i''1 ''i''2 ... ''i''''r''+1)(''j''1 ''j''2 ... ''j''''s''+1)...(''ℓ''1 ''ℓ''2 ... ''ℓ''''u''+1) be the unique decomposition of ''σ'' into disjoint cycles, which can be composed in any order because they commute. A cycle involving points can always be obtained by composing ''k'' transpositions (2-cycles):
:
so call ''k'' the ''size'' of the cycle, and observe that, under this definition, transpositions are cycles of size 1. From a decomposition into ''m'' disjoint cycles we can obtain a decomposition of ''σ'' into transpositions, where ''k''''i'' is the size of the ''i''th cycle. The number is called the discriminant of ''σ'', and can also be computed as
:
if we take care to include the fixed points of ''σ'' as 1-cycles.
Suppose a transposition (''a'' ''b'') is applied after a permutation ''σ''. When ''a'' and ''b'' are in different cycles of ''σ'' then
:,
and if ''a'' and ''b'' are in the same cycle of ''σ'' then
:.
In either case, it can be seen that , so the parity of ''N''((''a'' ''b'')''σ'') will be different from the parity of ''N''(''σ'').
If is an arbitrary decomposition of a permutation ''σ'' into transpositions, by applying the ''r'' transpositions after ''t''2 after ... after ''t''''r'' after the identity (whose ''N'' is zero) observe that ''N''(''σ'') and ''r'' have the same parity. By defining the parity of ''σ'' as the parity of ''N''(''σ''), a permutation that has an even length decomposition is an even permutation and a permutation that has one odd length decomposition is an odd permutation.
; Remarks:
* A careful examination of the above argument shows that , and since any decomposition of ''σ'' into cycles whose sizes sum to ''r'' can be expressed as a composition of ''r'' transpositions, the number ''N''(''σ'') is the minimum possible sum of the sizes of the cycles in a decomposition of ''σ'', including the cases in which all cycles are transpositions.
* This proof does not introduce a (possibly arbitrary) order into the set of points on which ''σ'' acts.
Generalizations
Parity can be generalized to Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
s: one defines a length function
In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
Definition
A length function ''L'' : ''G'' → R+ on a group ''G'' is a function sat ...
ℓ(''v''), which depends on a choice of generators (for the symmetric group, adjacent transposition
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
s), and then the function gives a generalized sign map.
See also
* The fifteen puzzle
The 15 puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle having 15 square tiles numbered 1–15 in a frame that is 4 tiles high and 4 tiles wide, leaving one unoccupied tile position ...
is a classic application
* Zolotarev's lemma
Notes
References
*
*
*
*
* {{cite book , last1=Meijer , first1=Paul Herman Ernst , last2=Bauer , first2=Edmond , title=Group theory: the application to quantum mechanics , series=Dover classics of science and mathematics , year=2004 , publisher=Dover Publications , isbn=978-0-486-43798-9
Group theory
Permutations
Parity (mathematics)
Articles containing proofs
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