let G be a group and let H be a subgroup of G

with |G : H | = 2.

(we know that if H is a subgroup of group G. Then (a) if H = {e} then |G: H | = | G | (b) if H = G then | G : H= 1 |

1. If K is a subgroup of G with at least one element not in H, show that G = HK.