How do you prove that an element and its inverse have the same order? I understand the idea but do not know how to prove it.
-Wooi-Chen
I thought this worked as a proof.
$ g^k=1 $ element g having order of k
$ (g^k)^{-1}=(1)^{-1} $
$ g^{-k}=1 $
$ (g^{-1})^k=1 $ inverse of g having order of k
This could be wrong, but it makes sense.
-Daniel
That actually makes sense to me as well. It is kind of playing with the order which power comes, that's the idea I get.
-Wooi-Chen
I think if you prove its cyclic the inverse will always be the same
-Matt
in this case, there are two cases where g has finite/infinite order of k.
- Panida
Daniel, have we defined what it is to take an inverse function? I am not sure if that is ok, to just take the inverse of both sides. And, can we bring the inverse in from the power? I am pretty sure it is ok to have the inverse of g^k is equal to the inverse of one, however I am not sure it is trivial to get that the inverse of g to the k is equal to one from this.
-Allen
