Let (G, *) be a group. Prove that the map π : G → G defined by π(g) = g * g is a homomoprhism if and only if G is abelian

Question

Let (G, *) be a group. Prove that the map π : G → G defined by π(g) = g * g is a

homomoprhism if and only if G is abelian

Answer 1

Suppose π is a homomorphism, and fix x, y ∈ G. Then (xy)² = x² y², or xyxy = xxyy. Left cancellation of x on both sides of the equation yields yxy = xyy. Right cancellation of y on both sides results in yx = xy. Since x and y were arbitrary, G is abelian,

Conversely, suppose G is abelian. Since xy = yx for all x, y ∈ G, we have

(xy)² = x(yx)y = x(xy)y = x^2 y^2

showing that π is a homomorphism.