Product of a subgroup with index 2 with another is the whole group Let $H,K\le G$ be two subgroups of $G$ with $K\not\le H$ and $[G:H]=2$. Then, is $HK=G$? I think yes, but am stammering in the proof. The index of $2$ suggests normality and hence, that maybe $HK$ is a subgroup. But, how could we show that $|HK|=|G|$? Thanks beforehand.

Since $K \
subseteq H$, there is $a \in K \setminus H$, and $aH \
e H$, so $HK \supseteq H \cup aH = G$.