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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, when ''X'' is a
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
with at least two elements, the
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

permutation
s of ''X'' (i.e. the
bijective function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 order, simple order, linear order, connex order, or full order is a binary relation on some Set (mathematics), set X, which is Antisymmetric relation, antisymmetric, Transitive relation, transitive, and a connex relation. ...
of ''X'' is fixed, the parity (oddness or evenness) of a permutation \sigma of ''X'' can be defined as the parity of the number of
inversions
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(s) 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 Theophrastus M ...
of the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
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 represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the natural numbers , for som ...
(''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), ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and mus ...

inversion
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
.Jacobson (2009), p. 50.


Example

Consider the permutation ''σ'' of the set which turns the initial arrangement 12345 into 34521. It can be obtained by three transpositions: first exchange the numbers 2 and 4, then exchange 1 and 5, and finally exchange 3 and 5. This shows that the given permutation ''σ'' is odd. Following the method of the Permutation#Cycle_notation, cycle notation article, this could be written, composing from left to right, as : \sigma=\begin1&2&3&4&5\\ 3&4&5&2&1\end = \begin1&3&5\end \begin2&4\end = \begin1&3\end \begin3&5\end \begin2&4\end . There are many other ways of writing ''σ'' as a functional composition, composition 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 and odd numbers, even number and only an even number of exchanges (called transposition (mathematics), 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
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. Furthermore, we see that the even permutations form a subgroup of S''n''. This is the alternating group on ''n'' letters, denoted by A''n''.Jacobson (2009), p. 51. It is the Kernel (algebra), kernel 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 of A''n'' (in S''n'').Meijer & Bauer (2004), [ p. 72] If , then there are just as many even permutations in S''n'' as there are odd ones; consequently, A''n'' contains factorial, ''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 cyclic permutation, cycle is even if and only if its length is odd. This follows from formulas like :(a\ b\ c\ d\ e)=(d\ e)(c\ e)(b\ e)(a\ e)\text(a\ b)(b\ c)(c\ d)(d\ e). 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 (group theory), 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 n points is also encoded in its cyclic permutation, 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 cycle notation, 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): :(a\ b\ c \dots x\ y\ z)=(a\ b)(b\ c) \dots (x\ y)(y\ z), 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 :n \text \sigma 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 :(a\ b)(a\ c_1\ c_2 \dots c_r)(b\ d_1\ d_2 \dots d_s) = (a\ c_1\ c_2 \dots c_r\ b\ d_1\ d_2 \dots d_s), and if ''a'' and ''b'' are in the same cycle of ''σ'' then :(a\ b)(a c_1 c_2 \dots c_r\ b\ d_1\ d_2 \dots d_s) = (a\ c_1\ c_2 \dots c_r)(b\ d_1\ d_2 \dots d_s). 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 t_1 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 groups: one defines a length function ℓ(''v''), which depends on a choice of generators (for the symmetric group, adjacent transpositions), and then the function gives a generalized sign map.


See also

* The fifteen puzzle 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 ru:Перестановка#Связанные определения