# Math Help - permutations, cosets, and direct products

1. ## permutations, cosets, and direct products

Can anyone help me with this problem please... thank you

Show that for every subgroup H of Sn for n ≥ 2, either all the permutations in H are even or exactly half of them are even.

2. I guess this approach might work:

Let $X$ be the set of all even permutations of $H$. Now if $X = H$ the proof is complete. Otherwise there exists a non-empty set of odd permutations $Y$.

Now you define a one-to-one onto map $\phi : X \mapsto Y$.
Define $\phi (\sigma) = (1,2)\sigma$ for $\sigma \in X$.

This shows that $|X| = |Y|$ thus there are exactly one half even permutations and one half odd permutations.