Question:

Herr Doktor Krebs:

We were working on Example 9.9, page 90 of the text and we came up with the following questions:

We were attempting to write the product of (1, 4, 5, 6)  (2, 1, 5) and express it as a product of transpositions, and we would like to know if all the following results are correct:

(1, 4, 5, 6)  (2, 1, 5) = (1, 6) (2, 4, 5) = (1, 4) (1, 5) (1, 6) (2,1) (2, 5) =

                                = (1, 4) (4, 5) (5, 6) (2,1) (1, 5) = (1, 6) (2, 4) (4,5)

Therefore there is an odd number of transposations but the number of transpositions can be different ( 3 or 5 in this case)?

Answer:

Your computations seem to be correct.  You are also correct that there may be many different ways to express the same permutation as a product of transpositions.  I may be possible to express the same permutation as a product of three transpositions, or of five, or of seventeen.  But if it's odd, it's odd, and if it's even, it's even.

-Herr Doktor Krebs