Exercise 3.1.11

Example. We can write (12345) (15)(14)(13)(12) 01' (12345) (12)(23)(34)(45). To see that this works, you just have to see where both sides send x E {1, 2,3, 4, 5}. Make a table: =|_ (15) ((14) 13)(12) x (12)( 23)(34)( 45) It follows that (12345) is an even permutation. In general, a cycle of odd length is an even permutation. A Lemma 3.1.1 Every transposition can be written as a product of an odd number of adjacent transpositions, that is, those of the form (a a + 1). Instead of proving the lemma in general, we give an example. Example. (14): (12)(23)(34)(32)(21). A Exercise 3.1.11 Check the preceding example and then nd the analogous decomposition of (13) and (27) in 57. What is the general formula for (ab) with a