abstract algebra?

Question

prove that any subgroup H of G with index 2 is normal but not conversly

Answer 1

Suppose x is not in H. Since H has index 2, every element of G is in either H or xH. Similarly, H and Hx also describe a partition of G. Therefore, we must have xH = Hx, or xHx^-1 = H, so H is a normal subgroup.

The converse is easy: just think of any subgroup of an abelian group which does not have index 2.