What is the order of an element in a finite abelian group of odd order?

[arshavin]

Homework Statement

Let G be a finite abelian group of odd order. Prove that the product of all the elements
of G is the identity.

The Attempt at a Solution

easy to see the case when each element has inverse which is not itself.

 

[micromass]

Indeed, if every element has an inverse which is not itself, then this is true. So, what you need to do is actually show that no element has itself as inverse. So, assume that g has itself as inverse, then what is the order of g?