1. The problem statement, all variables and given/known data Let G be a finite abelian group with no elements of order 2 Show that the function φ: G-> G defined as φ(g) = g^2 for all g ∈G, is an isomorphism. 2. Relevant equations Abelian group means xy = yx for all x,y∈G Isomorphic if there exists a bijection ϒ: G_1 -> G_2 such that for all x,y ∈ G, ϒ(xy) = ϒ(x)ϒ(y) 3. The attempt at a solution We have a couple of main points. -Abelian -Definition of the function maps one element to that element squared. -G is finite with no elements of order 2. To prove that it is isomorphic, we use the definition, that there must exist a bijection from G1 to G1 such that for all xy we see ϒ(xy) = ϒ(x)ϒ(y) I'm not sure where to go from here. Do I just say for some x, y in G we have ϒ(xy) = ϒ(yx) = (xy)^2 = (yx)^2 or some form of this ?