“Parity of a permutation”的意思、由来-开放百科全书

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Parity of a permutation

释义

Properties
Equivalence of the two definitions
Proof 1 Proof 2 Proof 3 Proof 4
Other definitions and proofs
Generalizations
See also
Notes
References

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering 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 {{math|1={{mvar|x}}, {{mvar|y}}}} of X such that and .

The sign or 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 of the symmetric group S_n. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (), 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

{{math|1=sgn(σ) = (−1)^N(σ)}}{{inconsistent citations}}}}
{{cite book |last1=Rotman |first1=J.J. |authorlink1= |last2= |first2= |authorlink2= |title=An introduction to the theory of groups |url= |edition= |series=Graduate texts in mathematics |volume= |year=1995 |publisher=Springer-Verlag |location= |isbn=978-0-387-94285-8 }}
{{cite book |last1=Goodman |first1=Frederick M. |authorlink1= |last2= |first2= |authorlink2= |title=Algebra: Abstract and Concrete |url= |edition= |series= |volume= |year= |publisher= |location= |isbn=978-0-9799142-0-1 }}
{{cite book |last1=Meijer |first1=Paul Herman Ernst |authorlink1= |last2=Bauer |first2=Edmond |authorlink2= |title=Group theory: the application to quantum mechanics |url= |edition= |series=Dover classics of science and mathematics |volume= |year=2004 |publisher=Dover Publications |location= |isbn=978-0-486-43798-9 }}Перестановка#Связанные определения
4 : Group theory|Permutations|Parity (mathematics)|Articles containing proofs

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