Parity of A Permutation - Example

Example

Consider the permutation σ of the set {1, 2, 3, 4, 5}  which turns the initial arrangement 12345 into 34521. It can be obtained by three transpositions: first exchange the places of 1 and 3, then exchange the places of 2 and 4, and finally exchange the places of 1 and 5. This shows that the given permutation σ is odd. Using the notation explained in the permutation article, we can write

\sigma=\begin{pmatrix}1&2&3&4&5\\ 3&4&5&2&1\end{pmatrix} = \begin{pmatrix}1&3&5\end{pmatrix} \begin{pmatrix}2&4\end{pmatrix} = \begin{pmatrix}1&5\end{pmatrix} \begin{pmatrix}1&3\end{pmatrix} \begin{pmatrix}2&4\end{pmatrix}.

There are many other ways of writing σ as a composition of transpositions, for instance

σ = (2 3) (1 2) (2 4) (3 5) (4 5),

but it is impossible to write it as a product of an even number of transpositions.

Read more about this topic:  Parity Of A Permutation

Famous quotes containing the word example:

    “Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
    —Marcel Proust (1871–1922)