If A is a group in which every element other than the identity, I, has order 2, prove that A is Abelian. Hence show that if X and Y are distinct elements of A, neither being equal to the identity, then the set {I, X,Y ,XY } forms a subgroup of A.
Deduce that if B is a group of order 2p, with p a prime greater than 2, then B must contain an element of order p.