Let G be a group. x and y are elements of G. Prove that:

a. The inverse of xy is y^-1x^-1

b. The identity element, e, is unique

c. The inverse of any element x of G is unique

d. If xy = xz then y = z

e. If x^-1y^-1=y^-1x^-1 then xy = yx

f. If every element x of G satisfies x x = e, then for any two elements, x, y, of G, we have xy = yx

Note that e is an element of G such that ex = x. Also note that for all the above, the group operation is not necessarily multiplication.