Properties
The identity permutation is an even permutation. An even permutation can be obtained from the identity permutation by an even number of exchanges (called transpositions) of two elements, while an odd permutation can be obtained by 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 Sn of all permutations of the set {1, ..., n}, we can conclude that the map
- sgn: Sn → {−1, 1}
that assigns to every permutation its signature is a group homomorphism.
Furthermore, we see that the even permutations form a subgroup of Sn. This is the alternating group on n letters, denoted by An. 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 An (in Sn).
If n > 1 , then there are just as many even permutations in Sn as there are odd ones; consequently, An contains n!/2 permutations.
A 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)
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 same as the parity of the permutation.
Every permutation of odd order must be even. The permutation (12)(34) in A4 shows that the converse is not true in general.
Read more about this topic: Parity Of A Permutation
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)