Example 3.7.11. Pick a positive integer n 2 2 and consider the group S. We define An = {o ES, | o is an even permutation}. We will use the first theorem above to verify that An is a subgroup of S. First of all, the identity is defined to be an even permutation, so it is in H. Now suppose that o and a are two even permutations. We noted back in 3.3.16 that their product is even. So we have on is in H. Finally, suppose that o is in H. We must show that or is also in H. As o is in H, it is even and so we may write 0 = 0102 . . . 0k, where each o; is a transposition and k is even. Every transposition is its own inverse and, using 3.1.17, we have o- = (01 . . .0k)-1 = 0k . ..01 = 0K . . .01 and we have written or as a product of an even number of transpositions and so it is in An as desired. We have verified the three conditions of the first theorem above and so An is a subgroup of Sn.3. List all left [BEETS of the alternating Subgroup :1\" of the Symmetric group 5\". {See 3.111.} Given permutation (I I: H\