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    Group Proofs : Properties of Groups

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    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.

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    Solution Preview

    a. Since (xy)*(y^-1x^-1)=x*(y*y^-1)*x^-1=x*e*x^-1=1. Thus the inverse of xy is y^-1x^-1.
    b. Suppose we have two indentiies e and e'. Then we have e*e'=e and e*e'=e'. This implies that e=e'. So the identity e of G is ...

    Solution Summary

    Group properties are invesigated.