Why is $[S_n:K]=2$ and $A_nK=S_n$ a contradiction?

Question

I'm trying to prove that $A_n$ is the only subgroup of index $2$ in $S_n$ by a contradiction argument, which have led me to the equality

$$A_nK=S_n,\tag{$*$}$$

where $K$ is a subgroup of index $2$ in $S_n$, distinct from $ A_n$. Why is $(*)$ a contradiction (if it actually is)?

Answer 1

This reasoning is possible.

Since every cycle of odd length lies in $K$ (why?) and $(123)(234)=(12)(34)$ it follows that $A_n\leq K$.