Induced Group Product is Homomorphism iff Commutative

Theorem
Let $$\left({G, \circ}\right)$$ be a group.

Let $$\phi: G \times G \to G$$ be defined such that $$\forall a, b \in G: \phi \left({\left({a, b}\right)}\right) = a \circ b$$.

Then $$\left({G, \circ}\right)$$ is abelian iff $$\phi$$ is an homomorphism.

Proof
We have $$\left({g_1, h_1}\right) \circ \left({g_2, h_2}\right) = \left({g_1 \circ g_2, h_1 \circ h_2}\right)$$.


 * Suppose $$\phi$$ is a homomorphism. Then:

This follows whatever $$g_2$$ and $$h_1$$ are, and so $$\left({G, \circ}\right)$$ is abelian.


 * Now suppose that $$\left({G, \circ}\right)$$ is abelian:

\end {proof}