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 |

(use this proposition: Let H and K be subgroups of a group G, let x be an arbritary element of G. There are several ways to construct (possibly) new subgroups of G :

(c) The set HK= {hk hH,kK } called the product of H and K is not always a subgroup. In fact HK is a subgroup of G iff HK= KH)

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