prove that A_3 inside S_3 is normal.
Does anyone have a good idea of how to go about proving this?
I just used the fact that parity (even and oddness) is a homomorphism and respects products.
--Jzage 16:14, 18 February 2009 (UTC)
Here's an idea. I'll let you judge if it's good.
We know that A3 is all even permutations of S3, which means that |A3| is exactly half of |S3| since there are an equal number of even and odd permutations. We also know that A3 is a subgroup in S3.
Now refer back to the first problem from this assignment. Let H = A3, G = S3. By theorem 7.1, the index (distinct right/left cosets) of H in G is |H|/|G| = 2. We proved that any subgroup H in G with index = 2 is normal in G. Therefore, A3 is normal in S3.
--Ysuo
