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.
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.
since its index is 2.
since its index is 2.
.
By second isomorphism theorem, . Similary, .
It follows that , which is isomorphic to Klein-4 group. Thus is not cyclic.