Alternating group is the unique subgroup of index 2 in Sn?

[ToffeeC]

Where n >= 2.

Is this true or false? I only got so far:

If K is a subgroup of index 2, then it's normal. K is normal in Sn, so it's a union of conjugacy classes. Also, since |An K| = |An| |K| / |An intersection K| = 1/2n! * 1/2n! / |An intersection K| <= n!, then 1/4n! <= |An intersection K|.

I don't know how to come up with a counter-example or proof from there. Could anybody help?

 

[Werg22]

An is simple. Try using that.