1. Suppose that H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of G of index 4 and that G/(H intersect K) is not cyclic. 2. Relevant equations- the back of my book says to use the Second Isomorphism Theorem for the first part which is... If K is a subgroup of G and N is a normal subgroup of G, then K/(K intersect N) is isomorphic to KN/N 3. The attempt at a solution- I know that any subgroup with order 2 is normal and that the intersection of 2 normal subgroups is normal. I just wasn't sure how to show it was index 4. Also, I wasn't what to do for the second part.