Suppose H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of index 4 and that G/(H intersect K) is not cyclic.
$\displaystyle \frac{H}{H \cap K} \triangleleft \frac{G}{H \cap K}$ since its index is 2.
$\displaystyle \frac{K}{H \cap K} \triangleleft \frac{G}{H \cap K}$ since its index is 2.
$\displaystyle \frac{H}{H \cap K} \cap \frac{K}{H \cap K}=1$.
By second isomorphism theorem, $\displaystyle \frac{H}{H \cap K} \cong \frac{HK}{K} \cong \frac{G}{K} \cong \mathbb{Z}_2$. Similary, $\displaystyle \frac{K}{H \cap K} \cong \mathbb{Z}_2$.
It follows that $\displaystyle \frac{G}{H \cap K} \cong \frac{H}{H \cap K} \oplus \frac{K}{H \cap K} \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2 $, which is isomorphic to Klein-4 group. Thus $\displaystyle \frac{G}{H \cap K}$ is not cyclic.