Alternating groups, specifically $A_6$ My question is what are the possible order of $A_6$? And how would I show I get $\frac{6!}{2}=360$. Any tips? I know that $A_6$ is the group of even permutations on six elements.  I also know that $360=2^3*3^2*5^1$.  How do I got about doing this?

Consider a homomorphism $S_n \rightarrow \{-1, 1\}$ defined such that $\phi(\pi) \mapsto \operatorname{sgn}(\pi)$.  
What is the kernel of this homomorphism?  From here, apply the fact that $S_n/\ker(\phi) \cong \operatorname{Im}(\phi)$.  What must be the orders of each of these?
Alternately, if you've proven that the only normal subgroup of $S_n$ is $A_n$, then what do you know about $\ker(\phi)$?
