Any thoughts?

We gave the deﬁnition of a group G under a binary operation. We say a group G is abelian if and and only if the binary operation for G is commutative; that is, g*h = h*g for all g and h in G. For an integer r > 0, write g*g*...*g=g^r.

Suppose an abelian group G has n elements. For any g in G, prove that g^n = e, where e is the identity of G. Do not use Lagrange’s Theorem, or any other theorem, from abstract algebra! [Follow the reasoning of the proofs of Fermat’s Theorem and Euler’s Theorem.]