Normal Subgroups of index 2

Question

Let $H$ and $K$ be distinct subgroups of $G$ with $[G:H]=2=[G:K]$.

$(i)$ Show that there exist a third subgroup $L$ of $G$ such that $[G:L]=2$

$(ii)$ Express $L$ in terms $G$,$H$ and $K$.

I tried to use second isomorphism theorem on (i) but i always get an index $4$. Any help and hints would be appreciated!!!

Answer 1

Note that since $H\triangleleft G$ and $K\triangleleft G$, then $HK=KH=G$. Thus, $[K\colon K\cap H] = [KH\colon H] = [G\colon H] = 2$, and similarly, $[H\colon K\cap H]=2$. As $K$ and $H$ are normal, $(K\cap H)\triangleleft G$, and $[G:K\cap H] = 4$.

Thus, $G/(K\cap H)$ is a group of order $4$. There are only two groups of order $4$. Only one of them as two distinct subgroups of order $2$.