It's really nice to read your website about the proof of A_n has index 2 in the function-theory; I found a simple typo in alternating group has index 2 in the symmetric group.
In short, it should be \pi(\lambda) = \alpha rather than \pi(\alpha) = \lambda.
For every element \alpha in A_n, there is an element \lambda in S_n - A_n, which is exactly \lambda = \pi^{-1} \alpha, to make \pi(\lambda) = \pi(\pi \alpha) = \alpha holds.
It's really nice to read your website about the proof of A_n has index 2 in the function-theory; I found a simple typo in alternating group has index 2 in the symmetric group.
In short, it should be \pi(\lambda) = \alpha rather than \pi(\alpha) = \lambda.
For every element \alpha in A_n, there is an element \lambda in S_n - A_n, which is exactly \lambda = \pi^{-1} \alpha, to make \pi(\lambda) = \pi(\pi \alpha) = \alpha holds.