Example
Consider the permutation ''σ'' of the set defined by and In one-line notation, this permutation is denoted 34521. It can be obtained from theProperties
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 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 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/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 correspondingEquivalence 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 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
* TheNotes
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