Let G be an abelian group and let .

If is finite then prove that, for , iff .

Now the obvious cases are if , and trivially. Same for .

The rest all seems rather straightforward, however I'm having trouble with finding a structure for my proof, especially since n is an integer, not just a natural number.

For the case where I have if then which, by definition of the order of an element cannot be equal to zero. I find it hard to prove this for the similar case of .

Am I making this too difficult or what is it that I'm missing?