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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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Group properties are invesigated.

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Proof:
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 ...

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