SUMMARY / RELATED TOPICS

In mathematics, when X is a finite set with at least two elements, the permutations of X fall into two classes of equal size: the permutations and the odd permutations. If any total ordering of X is fixed, the parity of a permutation σ 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 x < y and σ > σ. The sign, signature, or signum of a permutation σ is denoted sgn and defined as +1 if σ is and −1 if σ is odd; the signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol, defined for all maps from X to X, has value zero for non-bijective maps; the sign of a permutation can be explicitly expressed as sgn = Nwhere N is the number of inversions in σ. Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as sgn = mwhere 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. 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 exchange 1 and 3, exchange 1 and 5. This shows. Following the method of the cycle notation article, this could be written as σ = = =. There are many other ways of writing σ as a composition of transpositions, for instance σ =,but it is impossible to write it as a product of an number of transpositions; the identity permutation is an permutation. An permutation can be obtained as the composition of an number and only an number of exchanges 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 permutations is the composition of two odd permutations is the composition of an odd and an permutation is oddFrom these it follows that the inverse of every permutation is the inverse of every odd permutation is oddConsidering the symmetric group Sn of all permutations of the set, we can conclude that the map sgn: Sn → that assigns to every permutation its signature is a group homomorphism.

Furthermore, we see that the 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 but they form a coset of An. If n > 1 there are just as many permutations in Sn as there are odd ones. A cycle is if and only if its length is odd; this follows from formulas like = or. In practice, in order to determine whether a given permutation is 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 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. Every permutation can be produced by a sequence of transpositions: with the first transposition we put the first element of the permutation in its proper place, the second transposition puts the second element right etc.

Since we cannot be left with just a single element in an incorrect position, we must achieve the permutation with our last transposition. Given a permutation σ, we can write it as a product of transpositions in many different ways. We want to show that either all of those decompositions have an number of tran

Sistema Brasileiro de Televisão is a Brazilian free-to-air television network, funded on August 19, 1981 by the businessman and TV host Silvio Santos. The network was established after a public competition of the federal government for the creation of two new television networks, created from revoked concessions of the extinct networks Tupi and Excelsior. SBT was funded in the same day that the concession agreement was signed, that the act was broadcast live by the network, so that this was his first program aired. Is the second more watched television network in Brazil, after Rede Globo. Throughout its existence, the network always occupied this space in the audience ranking, except between 2007 and 2014, when Rede Record took the post. SBT has about 114 owned and operated station throughout the Brazilian territory, is available through pay television operators, by the free-to-air signal available in broadcast and satellite receivers, through streaming media in his mobile application, in apps for smart TVs and in his website.