Let have finite order , i.e. and is the smallest such exponents.
Thus,
Thus,
Thus,
Thus,
.
Now it remains to show that is smallest exponent for . Try showing that.
Prove that in any group, an element and its inverse has the same order.
My Proof:
Let G be a group, and suppose that and .
By definitions, such that and such that
thus,
Now, can I say that n must equal to m because the negative of inverse equals to the original power?