Suppose that $H$ and $K$ are two subgroups of a group $G$.Assume that $[G:H]=2$ and $K$ is not a subgroup of $H$.Show that $HK=G$.
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?