will someone please explain to me, for example in S_3, how do we find all the possible permutations, then from that how we find those permutations ( even ones) that are in A_3. I am looking at an example and it says that A_3 = {(1), (1,2,3), (1,3,2)}, but how can be (1,2,3), (1), and (1,3,2) be even permutations? Isn't the order of (1,2,3) = 3 ? I wrote down all the possible permutations of S_3, which are 6. Then I wrote them as cycles, so I got:

(1)(2)(3) , (1 2)(3), (1 3)(2), (2 3)(1), (1 2 3), (1 3 2). Can someone tell me exactly how I can find the order of a permutation, and how I can find the elements of A_3. I know that A_3 will have 3 elements, since its order is 3!/2 = 3.

The order of an element doesn't tell you whether it is even or odd.

A transposition has sign -1. The sign of a permutation is the product of the signs of any way of writing it as a a product of transpositions.

 

(123) = (12)(23)

 

so its sign is +1 as it is the product of an EVEN number of transpositions.

 

In general if we write an element as a product of disjoint cycles its sign is the product of the signs of the cycles. A cycle of length r has sign (-1)^{r+1}